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HolonomicFunctions (User's Guide)

1. np74 OreReduce S n Der z 3 Together gb n z na 4 2n a2 z 2 n 1 nz z 2 x3 SE g3 in275 OreReduce S n Der x 3 gb NormalizeCoefficients True owprsi nz a x 2 na x 2 in276 OreReduce S n Der z 3 gb NormalizeCoefficients True Extended True Out 274 P n l Out 276 ev 27 Ec 2 S nz a 2 nz az 6x n 1 S DS 2 2 x Dig 4 T E p gig 9n 20 n 1 n 1 n 1 2 n nz 10n 4x 4 24 oa n 2ng Un 2 HE 30 2 n d n 1 mn gt n 1 n 9n 20 4 n 2nz 11n 8a 28 3 naz 27 Br 42 2 en n 1 Sn n 1 Sch n 1 Sh n 3z 2 nx x 2 g n 1 ech n 1 H arri Inner 1 2 amp Last gb Plus 3 Out 277 085 b nz 2 z 2 5 nz x 2 SEE ALSO NormalizeCoefficients OreGroebnerBasis 71 OreSigma OreSigma op defines the endomorphism o for the Ore operator op MORE INFORMATION Ore operators like S or D are defined by two endomorphisms and A such that 6 is a o derivation i e that satisfies the skew Leibniz law f o f 9 g 9 f g Then the commutation rule for the newly introduced symbol is a c a 8 amp a The standard Ore operators shift differential delta Euler g shift are pre defined in HolonomicFunctions using OreSigma OreDelta and
2. Annihilator x n S n Der ax ouet105 Sh aD n 2x7 D 2nax 22 D n Ana n But it also works for inputs that do not form a Gr bner basis note that the elements of ann do not even belong to the same Ore algebra intoj ann Flatten Annihilator HermiteH n x amp S n Der a owpog S 21S 2n 2 D 2D 2n into7 DFiniteTimesHyper ann x n owpoz S 2278 2nz 227 D 2nz 22 D n Ana n Inf108j OreGroebnerBasis OreAlgebra S n Der x owpos S xD n 227 2 D 2nz 22 D n Ana n SEE ALSO Annihilator DFiniteOreAction DFinitePlus DFiniteSubstitute DFiniteTimes 30 DSolvePolynomial DSolvePolynomial leqn f E determines whether the ordinary linear differential equation eqn in ffx with polynomial coefficients has polynomial solutions and in the affir mative case computes them MORE INFORMATION The first argument egn can be given either as an equation with head Equal or as the left hand side expression which is then understood to be equal to zero If the coefficients of eqn are rational functions it is multiplied by their common denominator The second argument is the function to be solved for The algorithm works by determining a degree bound and then making an ansatz for the solution with undetermined coefficien
3. SEE ALSO Annihilator DFiniteOreAction DFinitePlus DFiniteTimes 28 DFiniteTimes DFiniteTimes ann anne computes an annihilating ideal for the product of the functions described by anni ann etc using the O finite closure property product MORE INFORMATION The ann are annihilating ideals given as lists of OrePolynomial expressions that form Gr bner bases this property is assumed and not checked separately The generators of all the ann have to be elements in the same Ore algebra and with the same monomial order Assume that ann annihilates the function fi then the output is an annihilating ideal for fi fo again given as a Gr bner basis The dimension of the vector space under the stairs of the output equals at most to the product of the vector space dimensions of the ann DFiniteTimes is called by Annihilator whenever a product of nontrivial ex pressions is encountered EXAMPLES npg annl Annihilator HermiteH n x S n Der x ou 9 Sn De 2x D 2x D 2n mpoj ann2 Annihilator r n S n Der x oupog Jr n S x npo DF initeTimes ann1 ann2 ourttoyj S 2D n 222 D 2na 23 D n Anz n host UnderTheStaircase Out 102 1 D mpo3 DFiniteTimes G Table ann2 10 Out 103 xD 10n S x SEE ALSO Annihilator DFiniteOreAction D
4. 13 0 ouj n 0 True n 6 True The following two recurrences annihilate 6 n all points on the diagonal are singular because of the leading coefficients and the point 0 0 is singular be cause usage of either recurrence would require values from outside the region Note that the cases returned by AnnihilatorSingularities may overlap as it appears here np ToOrePolynomial f n k 1 S k k n n k 1 S n k n oupsi k n 1 S k n k n 4 1 S k m npg AnnihilatorSingularities 0 0 oupa k gt n n gt 0 k gt 0 n 0 True L In the next example we have constant coefficients hence the only singularities correspond to values under the stairs Due to the additional restriction on the domain we get a parallelogram shaped set of initial values npsj ToOrePolynomial S k 2 1 S n 3 1 OreAlgebra S k S n Out 25 S2 4 3 1 nps AnnihilatorSingularities 1 1 Assumptions gt k lt n oups k 1 n gt 1 True k gt 1 n gt 2 True k gt 1 n 3 True k gt 2 n gt 2 True fk gt 2 n gt 3 True k gt 2 n 4 True SEE ALSO Annihilator AnnihilatorDimension UnderTheStaircase 10 ApplyOreOperator ApplyOreOperator opoly expr applies the Ore operator opoly to the expression expr MORE INFORMATION The first argument opoly can
5. k a k b 1 k a M inhomogeneous part Depending on whether the inhomogeneous part evaluates to zero or not we have P as an annihilating operator for the sum or we get an inhomogeneous relation for the sum In the latter case if one is not happy with that one can homogenize the relation by multiplying an annihilating operator for the 14 inhomogeneous part to P from the left Things become more complicated when the summation bounds involve the variables w since then additional correction terms have to be introduced the command Annihilator automatically deals with these issues Similarly we can derive relations for a definite integral Ip f z w dz In this case we look for creative telescoping operators that annihilate f and that are of the form T P w Ow D w Dr Ow Again it is straightforward to deduce a relation for the integral o Il b T x w Dz 8w f x w dx b P w 04 e f x w dx f D Uz w Dy Ow e f x w dx a b P w 0y e f x w da Q x w De 3w e f x w a which may be homogeneous or inhomogeneous The following options can be given Incomplete False If this option is set to True then the computation is stopped after the first creative telescoping relation is found makes sense only if ops contains more than one Ore operator Method Automatic The following methods can be chosen Chyzak Executes Chyzak s al
6. 64v 24v a n 8a n v 6a n 24a n 36a n v 11a n 32a ni 72a nv 44a nv Goin 16a v 48a v 44a v 12a v np ByteCount inhom oup2j 127664 np3j Simplify Simplify ReleaseHold inhom Limit myLimit A myLimit Limit Assumptions gt v gt 1 oun3j 0 0 0 OF mp4 rhs Annihilator 1 n Pi Gamma 2r n n Gamma v a 2b v Bessell v n ab E ab Der a Der b S n S v oupa an 2anv 2an 2av a S 2bvS bn 4bnv bn Ab 2bv D 2b VS 4 abn 4abnv abn Aabv 2abv n 4n v n An 2nv an 4anv an 4a 2av Da 2b vS abn Aabnv abn 4 4abv 2abv n 6n v n 12nv 4nv 8 Af 4b v Ab v S An v 20n v 24n v 32n T6n 44nv 16 56v 64v 24v S a n 8a n v 6a n 24a n 36a n v 11a n 32a n T2a n A4a nv 6a n 16a v 48a v 44a v 12a v his GBEqual Ihs rhs out 15 True SEE ALSO AnnihilatorDimension AnnihilatorSingularities CreativeTelescoping DFinitePlus DFiniteTimes DFiniteSubstitute DFiniteOreAction AnnihilatorDimension AnnihilatorDimension ann gives the dimension of the annihilating ideal ann MORE INFORMATION The input ann has to be a list of OrePoly
7. 52 n 1 3n 3n 1 S L n n 1 inzay 96 3 S n 2 n 2 owpa n 1 52 3n n 2 5 n 1 3n 3n 1 R n n n 1 aagi Exponent n Out 242 4 npa3j Variables p2 x Der a S n np44 MonomialOrder p1 Out 2 ER a out 244 DegreeLexicographic inp45 Normal p2 x Der r S n 1 Out 2 KS a SEE ALSO ApplyOreOperator ChangeMonomialOrder ChangeOreAlgebra LeadingCoefficient LeadingExponent LeadingPowerProduct LeadingTerm NormalizeCoefficients OreAlgebra OrePlus OrePower OreTimes OrePolynomialDegree OrePolynomialList Coefficients OrePolynomialSubstitute OrePolynomialZeroQ Support ToOrePolynomial 64 OrePolynomialDegree OrePolynomialDegree opoly gives the total degree of the Ore polynomial opoly with respect to all gen erators of the algebra OrePolynomialDegree opoly vars gives the total degree of opoly with respect to vars MORE INFORMATION The input opoly must be an OrePolynomial expression and vars a list of or a single indeterminates This list must either be a subset of the generators of the algebra or contain only elements that do not belong to the algebra EXAMPLES aagi opoly ToOrePolynomial n x S n Der a n 22 2 S n nz Der z 1 OreAlgebra S n Der z oupas n 2 Sn D n a 8 L nz D 1 npa7 OrePolynomialDegree
8. C 1 n 2 Nal gt C 1 n C 1 n n 2 in329 SolveCoupledSystem fl k 1 2 fl k x D f2 k x 3kv 2 x73 3 f1 k x x D f2 k x alt 1 f2 k x oupos f1 k 2 bei 2 k 2 gt ka CO SEE ALSO DSolveRational RSolveRational SolveOreSys 83 SolveOreSys SolveOreSys type var eqns fi var fi var pars computes all rational solutions of the first order coupled linear difference or differential system eqns MORE INFORMATION The input consists of five arguments type is either S or Der indicating that a difference resp differential system has to be solved var is the variable eqns is a list of equations fi var f var are the functions to be solved for and pars is a list of extra parameters that are allowed to occur linearly in the inhomogeneous parts The system of equations is uncoupled using the corresponding uncoupling pro cedure from Stefan Gerhold s OreSys package 7 this package has to be loaded in advance Then the scalar equations are solved with the functions DSolve Rational and RSolveRational respectively and by backwards substitution For the uncoupling some dummy functions psi are created they also show up in the output This command is kind of obsolete since SolveCoupledSystem offers more powerful solving abilities The following options can be given Method OreSys Zuercher
9. ChangeMonomialOrder ChangeOreAlgebra LeadingCoefficient LeadingExponent LeadingTerm NormalizeCoefficients OrePolynomial OrePolynomialList Coefficients ToOrePolynomial Support 47 Leading Term LeadingTerm opoly gives the leading term of the Ore polynomial opoly MORE INFORMATION The input opoly has to be an OrePolynomial expression What is considered as the leading term depends on the Ore algebra in which opoly is presented and of course on the monomial order By leading term we understand the lead ing power product or leading monomial multiplied by the leading coefficient The leading term of the zero polynomial is not defined Leading Term returns Indeterminate in this case EXAMPLES n165 opoly ToOrePolynomial 1 z 2 xx Der 2 x xx Der 4x 3 OreAlgebra Der z Out 166 1 x D xD Az npe7 Leading Term opoly oupez 1 x D np168j opoly ChangeOreAlgebra opoly OreAlgebra z Der x Out 168 r D 42 xD D np69 Leading Term opoly out 169 D npro Annihilator StruveH n x S n Der a oup 2 D 2nz x S 2nzD n n 4 2 z D n 1 S x Qna 3x S 4n 10n 2 6 S 2 D 3nz 3x ni7 LeadingTerm ounn x D2 xS Dr 2nz 4 31 92Y SEE ALSO ChangeMonomialOrder ChangeOreAlgebra LeadingCoefficient Leadin
10. lt S and then with S lt D pust gb Annihilator nSin zHarmonicNumber n 1 S n Der owing n 2n S D 2n 3n S n n xD na Sn Dr D NSn nzD n nS D n 1 Dz hagi FGLM gb Lexicographic ouma D DZ nzS D nS xD D nzD n n 2n S 2n 3n S D n n hani FGLM gb OreAlgebra Der z S n Lexicographic owpog n 3 S 3n 7 8 3n 5 S n 1 D S 2 D Sa Dr S 28 1 D n 2n S 2n 3n Sa n n 34 In the next example we compute in a module whose elements have 2 entries The two positions are indicated by the position variables pp and p Among the generators of the Ore algebra they sit on position 1 and 2 np21 ToOrePolynomial n po 2p1 2 2n p1 n p S n OreAlgebra po p S n o pzj npo 2p1 np Sn 2n 2 pi np2 FGLM OreAlgebra p1 po S n Lexicographic ModuleBasis 1 2 owp23 PoSn 2po 2p1 npo SEE ALSO AnnihilatorDimension OreGroebnerBasis GBEqual 35 FindCreativeTelescoping FindCreativeTelescoping expr deltas ops finds creative telescoping relations for expr by making an ansatz with ex plicit heuristically determined denominators in the delta parts With this command multiple summations and integrations can be done in one step by giving several deltas FindCre
11. 2nz D 4 n a Hn Support opoly 85 Sn Des Di Sn De 1 opoly ChangeOreAlgebra opoly OreAlgebra n x S n Dereli n z 2nxS 2nz D xS S 25 D D2 n Support opoly 1n 2 nz S ne Da SZ Sa Dr D2 ny Support opoly opoly SE E ALSO FindRelation LeadingPowerProduct OrePolynomial OrePolynomialList Coefficients 86 Takayama Takayama ann vars performs Takayama s algorithm for definite summation and integration with natural boundaries on a function annihilated by ann summing and integrat ing with respect to vars Takayama ann vars al 9 converts all elements of ann into the Ore algebra alg and then performs Takayama s algorithm MORE INFORMATION The input ann is a list of OrePolynomial expressions or a list of standard Mathematica polynomials then the third argument an Ore algebra into which these are translated has to be given vars is the list of variables with respect to which the summation and integration is done More generally vars contains all variables to be eliminated Takayama s algorithm is based on solving an elimination problem 11 The variables for which a shift operator belongs to the Ore algebra are inter preted as summation variables The variables for which a differential operator belongs to the Ore algebra are interpreted as integration variables The algorithm loops over the degree of vars
12. 79 RSolveRational RSolveRational eqn f 2 determines whether the linear recurrence equation egn in f n with poly nomial coefficients has rational solutions and in the affirmative case com putes them MORE INFORMATION The first argument egn can be given either as an equation with head Equal or as the left hand side expression which is then understood to be equal to zero If the coefficients of eqn are rational functions it is multiplied by their common denominator The second argument is the function to be solved for Following Abramov s algorithm 2 first the denominator of the solution is determined Then RSolvePolynomial is called to find the numerator polynomial The command RSolveRational is able to deal with parameters these have to occur linearly in the inhomogeneous part Call the parameterized version using the option ExtraParameters The following options can be given ExtraParameters specifies some extra parameters for which the equation has to be solved EXAMPLES ia RSolveRational 2n 7 f n 2 2n 2 7n 5 f n 1 6n 2 7n 3 f n 3 2n f n Out 317 In n4 zc r ngis eqn k 1 n n k f 1 k k 1 k n 1 f k k 1 k 1 ny 0 k 1 n 1 z 1 ou is k n 1 k n f k 1 k 1 k n 1 f k k 1 k n 1 0 k 4 1 n 1 f1 ag
13. Ip ED och 6 8 3 np Inner 1 xx 2 amp cofs x 3 Der x 2 Plus out 214j 1 SEE ALSO FGLM FindRelation GBEqual OreAlgebra OrePolynomial OreReduce Printlevel Under TheStaircase 59 OreOperatorQ OreOperatorQ expr tests whether expr is an Ore operator or not MORE INFORMATION This command gives True if expr is an Ore operator this is the case when the two endomorphisms OreSigmaf expr and OreDelta expr are defined Other wise False is returned Note that by the term Ore operator we mean a single symbol that has been introduced by an Ore extension In particular we do not mean an operator in the sense of a recurrence or a differential equation those are Ore polynomials involving some Ore operators EXAMPLES inpisj OreOperatorQ Der a oupisj True agi OreOperatorQ 0 o pig False npin OreSigma 0 e OreDelta 0 6 inzis OreOperatorQ 0 oup1sj True n19 ToOrePolynomial 0 xx a Out 219 ce a 8 a np2j OreOperatorQ ou o False SEE ALSO OreDelta OreOperators OreSigma 60 OreOperators OreOperators expr gives a list of Ore operators that are contained in ezpr MORE INFORMATION Every expression e for which the two endomorphisms OreSigma e and Ore Delta e are defined is considered as an Ore operator Note that by the term Ore operato
14. Sn n 1 1 D 2aD n n npo4j Printlevel 2 OreGroebnerBasis ann OreAlgebra S n Der z OreGroebnerBasis Number of pairs 1 OreGroebnerBasis Taking 4 2 2 1 2 OreGroebnerBasis Does not reduce to zero gt number 3 in the basis OreGroebnerBasis OreGroebnerBasis OreGroebnerBasis OreGroebnerBasis OreGroebnerBasis OreGroebnerBasis OreGroebnerBasis OreGroebnerBasis OreGroebnerBasis OreGroebnerBasis OreGroebnerBasis The Ipp is 1 1 The ByteCount is 1316 Number of pairs 2 Taking 5 2 1 1 3 Does not reduce to zero gt number 4 in the basis The Ipp is 1 0 The ByteCount is 1300 Number of pairs 3 Taking 5 1 21 2 3 Number of pairs 2 Taking 6 2 O 1 4 Number of pairs 1 Taking 6 1 1 3 4 Reducing no 1 of 2 Reducing no 2 of 2 DS 14 4 x 1 22 D2 2x D n n nposj Printlevel Infinity Annihilator Sum Binomial n k k 0 n S n Entering Annihilator Sum Annihilator called with Binomial n k Annihilator The factors that contain not to be evaluated elements are Annihilator The remaining factors are Gamma 1 k 1 Gamma 1 n Gamma 1 k n 1 Annihilator Factors that are not hypergeometric and hyperexponential CreativeTelescoping Trying d 0 ansatz eta 0 1 S k phi 1 k LocalOreReduce Reducing 0 0 LocalOreReduce Reducing 1 0 Start to solve scalar equat
15. and in the affirmative case computes them p 77 RandomPolynomial var deg c gives a dense random polynomial in the vari ables var of degree deg and with integer coefficients between c and c p 78 RSolvePolynomial leqn f n determines whether the linear recurrence equa tion eqn in f n with polynomial coefficients has polynomial solutions and in the affirmative case computes them p 79 RSolveRational eqn fin determines whether the linear recurrence equation eqn in f n with polynomial coefficients has rational solutions and in the affirmative case computes them p 80 S n represents the forward shift operator w r t n p 81 SolveCoupledSystem eqns Ip ful V1 v3 computes all rational so lutions of a coupled system of linear difference and differential equations p 82 SolveOreSys type var eqns fi var fx var pars computes all rational solutions of a first order coupled linear difference or differential system p 84 Support opoly gives the support of the OrePolynomial opoly p 86 Takayama ann vars performs Takayama s algorithm for definite summation and integration with natural boundaries p 87 ToOrePolynomial expr alg converts erpr to an Ore polynomial in the Ore algebra alg p 90 UnderTheStaircase gb computes the list of monomials power products that lie under the stairs of the Gr bner basis gb p 92 Annihilator Annihilator
16. bner bases are the same p 43 HermiteTelescoping expr Der y Derri performs Hermite telescoping to find creative telescoping relations for the hyperexponential expr p 44 LeadingCoefficient opoly gives the leading coefficient of the Ore polynomial opoly p 45 LeadingExponent opoly gives the exponents of the leading term of the Ore polynomial opoly p 46 LeadingPowerProduct opoly gives the leading power product of the Ore polynomial opoly p 47 LeadingTerm opoly gives the leading term of opoly p 48 NormalizeCoefficients opoly removes the content of the Ore polynomial opoly p 49 OreAction op determines how the newly defined Ore operator op acts on arbitrary expressions p 50 OreAlgebra gi 92 e d defines an Ore algebra that is generated by o g2 p 51 OreAlgebraGenerators alg gives the list of generators of the Ore algebra alg p 53 OreAlgebraOperators alg gives the list of Ore operators that are contained in the Ore algebra alg p 54 OreAlgebraPolynomialVariables alg gives the list of variables that occur polynomially in the Ore algebra alg p 55 OreDelta op defines the endomorphism for the Ore operator op p 56 OreGroebnerBasis Up e Pi computes a left Gr bner basis of the ideal generated by the Ore polynomials Pi PX p 57 OreOperatorQ expr tests whether expr is an Ore operator p 60 OreOperators expr gives a list of Ore operators th
17. expr ops computes annihilating relations for expr w r t the Ore operator s ops Annihilator expr automatically tries to determine for which operators relations exist MORE INFORMATION The input ezpr can be any Mathematica expression but not a list of expres sions and ops can be either a list of operators or a single operator admissible operators are S Der Delta and QS The output consists of a list of OrePoly nomial expressions which form a Gr bner basis of an annihilating left ideal for expr It need not necessarily be the maximal annihilating ideal De the full annihilator but often it is in particular when expr is recognized to be O finite If the input is not recognized to be O finite some heuristics to find relations are applied but it may well be that some are missed The relations are com puted by recursively analyzing the structure of the input down to its atomic building blocks and then executing the O finite closure properties DFinitePlus DFiniteTimes DFiniteSubstitute and DFiniteOreAction To see a com plete list of mathematical functions that are recognized by Annihilator as atomic building blocks whether 0 finite or not type Annihilator Annihilator has the attribute HoldFirst to prevent Mathematica from doing any simplification on the input If expr contains the command D or ApplyOre Operator then the closure property DFiniteOreAction is performed which is more desirable comp
18. given as Ore polynomials and the other con taining the corresponding inhomogeneous parts Critical components like Limit expressions of the inhomogeneous parts are wrapped with Hold Method Automatic is passed when CreativeTelescoping is called internally see p 14 MonomialOrder DegreeLexicographic the monomial order w r t to which the output is given see OreGroeb nerBasis p 57 for a list of supported monomial orders EXAMPLES npj Annihilator Sin Sqrt r 2 all Der z Der y o ti Dz 2xD 40 4y D 2D 1 h Annihilator LegendreP n x owp n 1 S 1 2 D na x x 1 D2 24D n n ng Annihilator ArcSin Sqrt x 1 k S k Der x Annihilator nondf The expression ArcSin Sqrt 1 x k is not recognized to be O finite The result might not generate a zero dimensional ideal ours J Ax S2 D 4a 2 92 D k 3k 2 ial Annihilator Sum Binomial n k k 0 n S n oufaj Sn 2 npj Annihilator Integrate LegendreP 2k 1 z 2 2 x 1 1 S k Assumptions Element k Integers amp amp k gt 0 Out 5 S 1 h Annihilator Fibonacci n S n Head f onse f n f n 1 ia 2 0 Note the difference between the following two ways to compute a differential equation for J x The closure property DFiniteOreAction never increases the order whereas DFinitePl
19. ot 1 x D aD 423 pagi LeadingCoefficient opoly 2 150 1 151 opoly ChangeMonomialOrder opoly Lexicographic 151 Ap r D xD D 152 LeadingCoefficient opoly 152 4 SEE ALSO ChangeMonomialOrder ChangeOreAlgebra LeadingExponent LeadingPowerProduct LeadingTerm NormalizeCoefficients OrePolynomial OrePolynomialList Coefficients ToOrePolynomial Support 45 LeadingExponent LeadingExponent opoly gives the exponent vector of the leading term of the Ore polynomial opoly MORE INFORMATION The input opoly has to be an OrePolynomial expression What is considered as the leading term depends on the Ore algebra in which opoly is presented and of course on the monomial order To match the definition of the degree of the zero polynomial the leading exponent of the zero polynomial is a list of ov s its length corresponding to the number of variables generators of the Ore algebra EXAMPLES np153j opoly ToOrePolynomial 1 x72 xx Der 2 x xx Der 4x 3 OreAlgebra Der z Out 153 1 x D a2D Ax np54 LeadingExponent opoly out 154j 2 Inp155j opoly ChangeOreAlgebra opoly OreAlgebra z Der x Out 155 r D 423 xD D2 isel LeadingExponent opoly oupiss 2 2 nps7 opoly ToOrePolynomial a b Sial S b OreAlgebrala b S a S b out 157 Gb S Sb nps8j LeadingExpone
20. 4 n 2 in254 OrePolynomialList Coefficients opoly oupsa 1 k k n 2 n 1 k n 2 k n 1 k n 2 in2s5 opoly ChangeOreAlgebra opoly OreAlgebra k m n S k S m S n obs K Sm S knSnS knS 2kSm Se nS RRR k 2kn kS n 3nS 3k 3n 2 2 npss OrePolynomialList Coefficients opoly oupss 1 1 1 2 1 1 1 2 1 21 3 3 3 2 2 SEE ALSO ChangeMonomialOrder ChangeOreAlgebra LeadingCoefficient NormalizeCoefficients OrePolynomial ToOrePolynomial 66 OrePolynomialSubstitute OrePolynomialSubstitute opoly rules applies the substitutions rules to the Ore polynomial opoly MORE INFORMATION The input opoly has to be an OrePolynomial expression and rules a list of rules of the form a gt b or a b Never try to use Mathematica s ReplaceAll for substitutions on Ore poly nomials This may cause results that are not well formed OrePolynomial expressions as the last example demonstrates The following options can be given Algebra None the Ore algebra in which the output should be represented None means that the algebra of opoly is taken MonomialOrder None the monomial order in which the output should be represented None means that the monomial order of opoly is taken EXAMPLES npsz opoly ToOrePolynomial S n Der z nzS n Der z 1
21. Out 115 07 n15 ChangeOreAlgebra OreAlgebra Der z owpug z D 282 Dy 266x9 D 10504 D 17012 D 9662 D 1272 D xD ha ChangeOreAlgebra OreAlgebra Euler x 8 ou 17 H SEE ALSO ApplyOreOperator Delta Der OreAction OreDelta OreSigma QS S ToOrePolynomial 33 FGLM FGLM 90 order transforms the Grobner basis gb of a zero dimensional ideal into a Gr bner basis for this ideal with respect to order using the FGLM algorithm FGLM 90 alg order translates the input into the new Ore algebra alg and then performs the FGLM algorithm MORE INFORMATION The input gb must be a list of OrePolynomial expressions that form a Gr bner basis with respect to the monomial order that is attached to these Ore polyno mials the Gr bner basis property is not checked by FGLM Note that this implementation works only for zero dimensional ideals The following options can be given ModuleBasis when you deal with Gr bner bases over a module give here a list of natural numbers indicating the location of the position variables among the gener ators of the Ore algebra EXAMPLES We start with a left ideal that is computed by Annihilator hence it is a Gr bner basis with respect to degree order To extract the pure ODE resp re currence we use the FGLM algorithm to get a Gr bner basis with lexicographic order first with D
22. data contains a list of terms each term being a pair consisting of a coefficient and an exponent vector The exponents are given in the order as the generators of the algebra are given The zero polynomial has the empty list in this place The second argument alg is an Ore algebra i e an expression of type OreAlgebraObject The third argument contains the monomial order with respect to which the entries in data are ordered see the description of OreGroebnerBasis p 57 for a list of supported monomial orders Note that data is always kept ordered such that the leading term corresponds to the first list element In the following let p p and py be OrePolynomial expressions The algebra and the monomial order of an Ore polynomial can be extracted by OreAlge bra p and MonomialOrder p respectively For the type OrePolynomial numerous upvalues have been defined to make life easier First this concerns the arithmetical operations Plus Times Non CommutativeMultiply and Power This means that you can type p po instead of the more cumbersome OrePlus p1 pa and p1 p2 instead of Ore Times p1 p2 the symbol is Mathematica s notation for NonCommuta tiveMultiply Attention when using the commutative Times it should only be used in cases where noncommutativity does not play any role e g for 2 pi or p Otherwise it can happen that Mathematica reorders the factors and the output is not what you originally wanted Next th
23. function fi then the output is an annihilating ideal for f fo or more precisely for any linear combination c fi c f2 again given as a Gr bner basis The dimension of the vector space under the stairs of the output equals at most to the sum of the vector space dimensions of the ann Note also that the output corresponds to the intersection of the input ideals which is the left LCM in case of univariate recurrences or differential equations DFinitePlus is called by Annihilator whenever a sum of nontrivial expres sions is encountered The following options can be given MonomialOrder None specifies the monomial order in which the output should be given None means that the monomial order of the input is taken EXAMPLES miezh annl Annihilator HermiteH n x S n Der x ps S D 2x D 2x D 2n nssj Under TheStaircase ann1 oufes 1 De legi ann2 Annihilator r n S n Der ax ou eg xD n Sn x aral UnderTheStaircase ann2 ou zj 1 az DFinitePlus ann1 ann2 on na r D n n S n 2na n 2z D An x 2na 2nz n a S D na x S na x D 2n 2n a n S2 Anz 3a 2a S 2nz 23 2 D 2n 2nz 2n 2x 23 ny3 Under TheStaircase 96 oupzj 1 Dr Sn 22 The expression n 1 n n n is obviously anni
24. in ann whose coefficients are of a special form the variable that represents q occurs only with powers that are multiples of lem m kj and o itself occurs only with powers divisible by kj for all 1 j X d respectively Then in these recurrences the substitutions d gt e 7 m d 5 can be safely performed note that performing these substitutions in ann directly would in general lead out of the underlying Ore algebra Let u denote the number of monomials under the stairs of ann then the output will have at most w Ia kj lem m k 7 monomials under the stairs where N j is the number of operators of the form QS oe in the algebra The following options can be given ModuleBasis when dealing with annihilating modules give here a list of natural numbers indicating the location of the position variables among the generators of the Ore algebra Can be used for dealing with inhomogeneous recurrences Return Annihilator specifies the type of the output Annihilator By default a Grobner basis for the annihilating ideal of the resulting sequence is returned Backsubstitution Returns some elements of ann in whose coefficients the powers of q are multiples of lem m kj and the powers of d are divisible by kj Substituting ou gt e 7 m oi 5 in this result yields exactly the output of Return Annihilator Support Only the support of the annihilating ideal of the resulting se quence is return
25. opoly Out 247 2 aagi OrePolynomialDegree opoly S n Out 248 1 aagi OrePolynomialDegree opoly n cl Outpag 4 mpsoj opoly ChangeOreAlgebra opoly OreAlgebra S n Der z n zl oupso Bn e 28 nz S D n Drt 512 Dine SD 8 n 1 nps OrePolynomialDegree opoly Out 251 5 SEE ALSO LeadingExponent LeadingPowerProduct OrePolynomial Support ToOrePolynomial 65 OrePolynomialList Coefficients OrePolynomialList Coefficients opoly gives a list containing all coefficients of the Ore polynomial opoly ordered according to the monomial order MORE INFORMATION The output contains the coefficients as they appear in the OrePolynomial expression opoly In particular it does not contain zeros and hence is different from Mathematica s CoefficientList EXAMPLES nps Annihilator k n StirlingS1 k m StirlingS2 m n S k S m S n Annihilator nondf The expression StirlingS1 k m is not recognized to be O finite The result might not generate a zero dimensional ideal Annihilator nondf The expression StirlingS2 m n is not recognized to be O finite The result might not generate a zero dimensional ideal owpsaj Sk Sm Sn k kn 2k Sm Sn kn k n 3n 2 52 k 2kn 3k n 3n 2 w sst opoly Factor First out 253 SS S k k n 2 8 S n 1 kK n 2 S k n 1 k
26. or D are defined by two endomorphisms and A such that 6 is a o derivation i e that satisfies the skew Leibniz law f o f 9 g 9 f g Then the commutation rule for the newly introduced symbol is a c a 8 amp a The standard Ore operators shift differential delta Euler g shift are pre defined in HolonomicFunctions using OreSigma OreDelta and OreAc tion If you want to define your own Ore operators use OreSigma and Ore Delta to define their commutation properties Note that op can be a pattern as well as a fixed expression EXAMPLES npoo OreDelta Der a out 200 Ox HI amp npo OreDelta S n out 201 U amp We show how a generic Ore operator can be defined ipo3 OreSigma MyOp MySigma OreDelta MyOp MyDelta manat ToOrePolynomial MyOp 2 zg a ouf203 MySigma MySigma a MyOp MyDelta MySigma a MySigma MyDelta a MyOp MyDelta MyDelta a Now we introduce the double shift i e the shift by 2 inzoa OreSigma S2 a A a gt a 2 amp mpos OreDelta S2 0 amp mpos OreAction S2 a_ OreSigma S2 a npo7 ToOrePolynomial n S2 n 1 2 owporl n 2n 522 2nS2 1 inzos ApplyOreOperator f n oapos n 2n f n 4 4 f n 2nf n 2 SEE ALSO OreAction OreOperatorQ OreOperators OreSigma 56 OreGroebnerBasis OreGroebner Basis Up gel Bj computes a left Gr bner basis for the
27. specify some extra parameters for which the equation has to be solved these have to occur linearly in the inhomogeneous part EXAMPLES miu DSolveRational z 2 x f a2 x fiel oupui Uo m nu DSolveRational a 2 2a 3 a 4 f a 8a 21a 2 13a 3 f a 12 53a 44a 2 f a 27 36a f a f a 4a O 2 4aC 1 3C 1 2C 2 oan lau HOL these 2088 In 113 DSolveRational u 1 u f u u 1 u 1 2u 2au u 2u 2au f lu ut aut 3u 2au a u ER 4au 2a u u au u 2au a u9 f u w 1 u c 0 uc 1 v c 2 flu ExtraParameters c 0 ell c 2 Out 113 TUS N aC lu aC l u Cla C 1 u C 1 u C 1 u c 0 a 3a 3a 1 C 1 e 1 2 a 2a 1 C 1 S S gt a 4 1 zr SEE ALSO DSolvePolynomial QSolvePolynomial QSolveRational RSolvePolynomial RSolveRational 32 Euler Euler 2 represents the Euler operator 0 xD MORE INFORMATION When this operator occurs in an OrePolynomial object it is displayed as 0 The symbol Euler receives its meaning from the definitions of OreSigma OreDelta and OreAction Functions like Annihilator cannot deal with mcEuler use Der x instead to represent differential equations EXAMPLES np114 ToOrePolynomial Euler z Out 114 Or wgl 96 8
28. the involved Ore operators and all other generators of alg if specified in particular these are not allowed in the denominator Also all Ore operators that occur in expr must be part of alg If an equation eqn is given then the Ore operators are determined by the occurrences of f The equation eqn must be linear and homogeneous If expr is a standard Mathematica polynomial i e if it does not involve Non CommutativeMultiply then it is assumed that in its expanded form all monomials are in standard form according to the order of the generators of alg regardless how Mathematica sorts the factors Otherwise use NonCommutativeMultiply written as to fix the order of the factors in a product If no Ore algebra is given then the rational Ore algebra that is generated by all Ore operators in expr is chosen The following options can be given MonomialOrder DegreeLexicographic the monomial order in which the terms of the Ore polynomial are ordered see the description of OreGroebnerBasis p 57 for a list of supported monomial orders EXAMPLES aagi ToOrePolynomial 1 z 2 Der z n 1 S n x n owus 1 z n 1 8 7nz x n359 ToOrePolynomial az n f n 1 n f 1 n 1 2 D f x n e fl n owssg 1 z n 1 8 ne x nps ToOrePolynomial Der z x OreAlgebra x Der z Ou ss1 2D n353 ToOrePo
29. this option is passed to the uncoupling procedure EXAMPLES Inf330 lt lt OreSys m OreSys Package by Stefan Gerhold RISC Linz V 1 1 12 02 02 i331 SolveOreSys Der x 1 n f x x nx g a 1 2 2 g 0 1 z 2 x nx f x 1 n g C1 2 2 f z Lle allt Out 331 EE gt HolonomicFunctions Private psi 24336 2 x gt fle gt 2 ole 4M aad SolveOreSys S n 84 f 2 n a 1 n 24 n b n 3 2n b 1 n 0 2 n 2 n a n 1 n b 1 n 0 aln balt H Out 332 EE gt C 1 n C 1 n 4 2 1 1 2n n 2 HolonomicFunctions Private psi 24516 2 n gt OUt a E D SR a n gt C i n C 1 n 2 bn gt n 1 C 1 n 2 H SEE ALSO DSolveRational RSolveRational SolveCoupledSystem 85 Support Support opoly gives the support of the OrePolynomial opoly MORE INFORMATION The input has to be given as an OrePolynomial expression By its support we understand the list of power products which have nonzero coefficient These power products are returned again as OrePolynomial expressions EXAMPLES In 333 0ut 333 In 334 0ut 334 In 335 Out 335 In 336 Out 336 In 337 0ut 337 opoly ToOrePolynomial S n Der a nax 2 OreAlgebra S n Der z K S2 25 D D 2nz 2 Sn
30. trying to find an operator free of vars in the module that is truncated at this degree Takayama can run into an infinite loop for two reasons either the input is not holonomic or it is holonomic but this property is lost by extension contraction The fact which degrees are tried can be influenced by the options StartDegree and MaxDegree The following options can be given Extended False setting this option to True computes also the delta parts which can be very costly Incomplete False setting this option to True causes that the computation is interrupted as soon as an element free of the elimination variables vars is found Method sugar this option is passed to OreGroebnerBasis and specifies the pair selection strategy Modulus 0 give a prime number here for modular computations Reduce False if this is set to True then the delta parts are reduced to normal form with respect to the input annihilator works only in connection with Extended 87 Saturate False when this option is set to True then it is tried to saturate the annihilating ideal in the polynomial algebra Weyl closure by an additional Gr bner basis computation this can increase the number of solutions and or de crease the order of the result StartDegree 0 the degree with respect to vars from which on it is tried to find operators free of vars in the truncated module MaxDegree Infinity the
31. 9 Sergej A Abramov Rational solutions of linear difference and q difference equations with polynomial coefficients In Proceedings of the International Symposium on Symbolic and Algebraic Computation ISSAC pages 285 289 New York NY USA 1995 ACM Sergej A Abramov and Moulay Barkatou Rational solutions of first order linear difference systems In Proceedings of the International Symposium on Symbolic and Algebraic Computation ISSAC pages 124 131 New York NY USA 1998 ACM Moulay Barkatou On rational solutions of systems of linear differential equations Journal of Symbolic Computation 28 547 567 1999 Alin Bostan Shaoshi Chen Fr d ric Chyzak Ziming Li and Guoce Xin Hermite reduction and creative telescoping for hyperexponential functions In Proceedings of the International Symposium on Symbolic and Algebraic Computation ISSAC New York NY USA 2013 ACM To appear preprint on arXiv 1301 5038 Fr d ric Chyzak An extension of Zeilberger s fast algorithm to general holonomic functions Discrete Mathematics 217 1 3 115 134 2000 Stefan Gerhold Uncoupling systems of linear Ore operator equations Mas ter s thesis RISC Johannes Kepler University Linz 2002 Israil S Gradshteyn and Josif M Ryzhik Table of Integrals Series and Products Academic Press Elsevier 7th edition 2007 Alan Jeffrey and Daniel Zwillinger eds Christoph Koutschan Advanced applications of the holonomic syste
32. D n 2n 1 n63 Under TheStaircase 20 ou sz 1 De n53 DFinitePlus Annihilator LegendreP n 1 2 S n Der Annihilator LegendreP n x S n Der a oaiet 2n 2 Sn D 1 ax D 2na 4a Dy n 3n 2 2n 12n 22n 12 S 2 2x7 1 D 4n x 20n x 32nz 16x S 4 2na F2nz 4r dar D 2n n z 9n 3nz 13n 22 6 xf 2x 1 DI 423 4x D2 2n 8n 10n AIS 3n z 3n na In 22 2 D 2n x 8n x 10nz 4x n54 Under TheStaircase Out 64 1 Di Sh D Sometimes the dimension can even drop Inssj ann Annihilator HarmonicNumber n S n owjs n 2 2n 3 S n 1 n55 DFiniteOreAction ann S n 1 osi n 2 8 n 1 SEE ALSO Annihilator DFinitePlus DFiniteSubstitute DFiniteTimes 21 DFinitePlus DFinitePlus anni anng computes an annihilating ideal for the sum of the functions described by ann anna etc using the 0 finite closure property sum MORE INFORMATION The ann are annihilating ideals given as lists of OrePolynomial expressions that form Gr bner bases this property is assumed and not checked separately The generators of all the ann have to be elements in the same Ore algebra and with the same monomial order Assume that ann annihilates the
33. FinitePlus DFiniteSubstitute DFiniteTimesHyper 29 DFiniteTimesHyper DFiniteTimesHyper ann expr computes an annihilating ideal for the product f expr where f is annihilated by ann and expr is hypergeometric and hyperexponential in all variables under consideration MORE INFORMATION The input ann is a list of OrePolynomial expressions or a single such ex pression which for this special command need not form a Gr bner basis The expression expr has to be hypergeometric with respect to all discrete vari ables of ann and hyperexponential with respect to all continuous variables of ann Assume that ann annihilates some function f then the output operators annihilate the product f expr The result is obtained by simple rewrite rules namely to multiply to each term some rational function that is determined by the exponents of the shift and differential operators and that compensates the rational function that appears by shifting and differentiating expr In standard situations where ann is given as a Gr bner basis the command DFiniteTimes is preferable EXAMPLES Of course DFiniteTimesHyper works also when the input is a Gr bner basis Then it delivers the same output as DFiniteTimes npo4 DFiniteTimesHyper Annihilator HermiteH n x S n Der x r n oupos S xD n 222 D 2na 23 D n Anz n mposj DFiniteTimes Annihilator HermiteH n x S n Der
34. HolonomicFunctions User s Guide Christoph Koutschan Research Institute for Symbolic Computation RISC Johannes Kepler University A 4040 Linz Austria June 17 2013 This manual describes the functionality of the Mathematica package Holo nomicFunctions It is a very powerful tool for the work with special functions it can assist in solving summation and integration problems it can automati cally prove special function identities and much more The package has been developed in the frame of the PhD thesis 9 The whole theory and the al gorithms are described there and it contains also many references for further reading as well as some more advanced examples the examples in this manual are mostly of a very simple nature in order to illustrate clearly the use of the soft ware HolonomicFunctions is freely available from the RISC combinatorics software webpage www risc uni linz ac at research combinat software HolonomicFunctions Short references Annihilator expr ops computes annihilating operators for the expression expr with respect to the Ore operators ops p 5 AnnihilatorDimension ann gives the dimension of the annihilating left ideal ann p 8 AnnihilatorSingularities ann start computes the set of singular points for a system ann of multivariate recurrences p 9 ApplyOreOperator opoly expr applies the operator given by the Ore poly nomial opoly to expr p 11 ChangeMonomialOr
35. Order None if this option is used then also the monomial order is changed None means that the monomial order of the input is taken EXAMPLES nssj opoly ToOrePolynomial 2 3 2 2 x 1 Der a 2 62 2 Az 2 xx Der 62 2 OreAlgebra Der z oups 2 2 1 D2 62 4x 2 D 6x 2 ael ChangeOreAlgebra opoly OreAlgebra z Der z oups 2 D 2 D 62 D zD 4zD D 6x 2D 2 azh ChangeOreAlgebra opoly OreAlgebra Der x zl wien D2a D2x Dr D agl ChangeOreAlgebra opoly OreAlgebra S n ChangeOreAlgebra ops Some of the operators Der x occur in the polynomial but are not part of the algebra outf38j Failed mpo 1 a opoly 1 2 2 owisj x OG 1 D2 6a TUD 8 ail ChangeOreAlgebra OreAlgebra z Der z ChangeOreAlgebra nopoly The elements of the new OreAlgebra do not occur polynomially outfaoj Failed SEE ALSO ChangeMonomialOrder OreAlgebra OrePolynomial ToOrePolynomial 13 CreativeTelescoping CreativeTelescoping expr delta ops finds creative telescoping relations for expr with Chyzak s algorithm i e operators of the form P 4 delta Q such that P involves only Ore operators from ops and their variables CreativeTelescoping ann delta ops finds creative telescoping relations in the annihilating ideal ann MORE INFORMATION The first argume
36. OreAc tion If you want to define your own Ore operators use OreSigma and Ore Delta to define their commutation properties Note that op can be a pattern as well as a fixed expression EXAMPLES inz7j OreSigma Der z out 278 j 1 amp nprg OreSigma S n owr Z1 n gt nt 1 amp We show how a generic Ore operator can be defined npso OreSigma MyOp MySigma OreDelta MyOp MyDelta nps ToOrePolynomial MyOp 2 zg a oups MySigma MySigma a MyOp MyDelta MySigma a MySigma MyDelta a MyOp MyDelta MyDelta a Now we introduce the double shift i e the shift by 2 m2s2 OreSigma S2 a a a 2 amp mpssj OreDelta S2 0 amp nps4 OreAction S2 a OreSigma S2 a inpss ToOrePolynomial n S2 n 1 2 owpssi n 2n 522 2nS2 1 nps ApplyOreOperator f n oapss n 2n f n 4 4 f n 2nf n 2 SEE ALSO OreAction OreDelta OreOperatorQ OreOperators 72 OreTimes OreTimes opoly1 opoly2 computes the product of the Ore polynomials opolyi opolye etc OreTimes lopoly opoly2 alg translates opoly opoly2 etc into the Ore algebra alg and then computes their product MORE INFORMATION The input polynomials can be either given as OrePolynomial expressions or as standard Mathematica polynomials OreTimes then tries to figure out which Ore algebra is best suited for representing
37. OreAlgebra Der z S n oupsz Dr Sn D nzS 1 n255 OrePolynomialSubstitute opoly Der x 1 oupss nz 1 Sn 2 wesst OreAlgebra ouf259 K n z Ds 1 Dz Sn Sn 0 aisen OrePolynomialSubstitute opoly Der 1 Algebra OreAlgebra S n out 260j na 1 S 2 inz61 OrePolynomialSubstitute opoly n 0 oupei Dr Sn D 1 The following gives nonsense mpej opoly n 0 Out 262 D S D OS 1 SEE ALSO OrePolynomial ToOrePolynomial 67 OrePolynomialZeroQ OrePolynomialZeroQ opoly tests whether the Ore polynomial opoly is zero MORE INFORMATION Note that this command does not do any simplification on the coefficients EXAMPLES n263 opoly ToOrePolynomial nS n S n OreAlgebra S n Out 263 n 1 5 n 264 OrePolynomialZeroQ opoly out 264 False zegi OrePolynomialZeroQ opoly S n n out 265 True zegi opoly ToOrePolynomial Der 1 Sin 2 Cos a 2 OreAlgebra Der z owpsg D 1 Cos z Sin z nps7 OrePolynomialZeroQ opoly Der a Oout 267 False SEE ALSO OrePolynomial ToOrePolynomial 68 OrePower OrePower opoly n computes the n th power of the Ore polynomial opoly MORE INFORMATION The input opoly is an OrePolynomial expression and n has to be an integer Negative integers are only
38. S n 2n 1 np74 NormalizeCoefficients opoly Extended True Out 172 n n S Sn n 1 oaa Bn n Se 4 S G 2 ex SEE ALSO OrePolynomialList Coefficients ToOrePolynomial 49 OreAction OreAction op defines how the Ore operator op acts on arbitrary expressions MORE INFORMATION An Ore operator like S or D in the first place is defined by the two endomor phisms o and 6 But also the action of this operator has to be fixed usually it is either equal to o e g in the case of a shift operator or to 6 e g in the case of a differential operator The definition of OreAction is used by ApplyOreOperator The standard Ore operators shift differential delta Euler g shift are pre defined in HolonomicFunctions using OreSigma OreDelta and OreAc tion If you want to define your own Ore operators use OreAction to define how they should act on expressions Note that op can be a pattern as well as a fixed expression EXAMPLES m5 OreAction Der z out 175 Or HI amp np7e OreSigma MyOp MySigma n7 OreDelta MyOp MyDelta m78 OreAction MyOp NY Action np7g ToOrePolynomial MyOp 2 Out 179 MyOp nusq ApplyOreOperator a 1 owe MyAction MyAction a 1 Now we introduce the double shift i e the shift by 2 nps1 OreSigma S2 a A a gt a 2 amp nps OreDelta S2 0 a
39. a 58 Modulus 0 give a prime number here for modular computations NormalizeCoefficients gt True whether the content of intermediate and final results shall be removed Reduce True whether autoreduction should be performed at the end of Buchberger s al gorithm ReduceTail True whether also the tail i e the terms that come behind the first non reducible term should be reduced for each S polynomial EXAMPLES znal rels Flatten Annihilator GegenbauerC n m x amp S n S m Der x out 209j n 2 82 2ma 2nz 2x S 2m n Am a 4m Ama 4m S L CAm a 8m Amna 4mn 4m2 6m S 4m 4mn 2m n n a 1 D 2ma x D 2mn n inz19 OreGroebnerBasis rels OreAlgebra S m S n Der oupi n 1 S 1 D 2mz na 2m5m xD 4 1 32 D 2mz z D 2mn By Gr bner basis computation we can show that relations are not compatible inzuj opolys ToOrePolynomial S k 2 n k S k n 2 k 2 S n 2 2n S n ky OreAlgebra S Kk S n owpix Sg n k Se n EL S2 2n8 k i212 OreGroebnerBasis opolys oup 1 We demonstrate the use of the option Extended mpi3 gb cofs OreGroebnerBasis 2 3 Der v 2 OreAlgebra Der z Extended True Out 213 ax Zu p
40. a homomorphic image and then the solution is computed in the original domain FindSupport only modular computations are done and the minimal ansatz is returned as a list of options that can be given to FindCre ativeTelescoping again in order to perform the final computation or a modular one Modular this mode requires to specify an ansatz in the way as it is returned by Mode FindSupport and is supposed to be used to gether with the options OrePolynomialSubstitute Modulus and FileNames 36 Support Automatic Specify the support of the principal part By default FindCreativeTele scoping loops over the support until a set of principal parts is found that generates a zero dimensional ideal Degree Automatic Specify the degree of the summation and integration variables to be used in the numerators of the delta parts By default FindCreativeTelescoping loops over the degree up to a heuristically determined degree bound If the support is given with the option Support then it loops ad infinitum if no creative telescoping operator is found Denominator Automatic By default the denominators of the delta parts are determined automatically Setting Denominator gt d for a polynomial d uses d as denominator in each delta part Setting Denominator di1 din dmi dmn uses dj as the denominator of the j th term in the i th delta part hence n is the number of monomials under the stairs of the i
41. admissible in degenerate cases when opoly does not contain any Ore operators To make the work with Ore polynomials more convenient we have defined an upvalue for Power This means that the standard notation opoly n can be used and OrePower will be called automatically EXAMPLES zegi opoly ToOrePolynomial n S n Der z Out 268 NIX Dr Sn zegi OrePower opoly 3 ourpes n z 3n 2 2na DFS 3n z 9n z 6nax D S na 3n7a 2nz DS mpzoj 1 opoly OrePower negpow Negative power of an OrePolynomial oup70 Failed n71 OrePolynomialSubstitute opoly S n 1 Der x 1 1 Out 271 n SEE ALSO OrePolynomial OrePolynomialSubstitute OrePlus OreTimes ToOrePolynomial 69 OreReduce OreReduce opoly Jo os D reduces the Ore polynomial opoly modulo the set of Ore polynomials 915 92 Ser T MORE INFORMATION The g are OrePolynomial expressions and opoly may be either an OrePoly nomial expression or a standard Mathematica polynomial in the latter case it is translated into the Ore algebra in which the g are given Note that the set 91 os needs not to form a Gr bner basis However if it is not the result of the reduction may not be uniquely defined As always the operations in OreReduce involve only multiplications from the left By default no content is removed during the reduction Thus denominators may appe
42. and in the affirmative case computes them MORE INFORMATION The first argument eqn can be given either as an equation with head Equal or as the left hand side expression which is then understood to be equal to zero If the coefficients of eqn are rational functions it is multiplied by their common denominator The second argument is the function to be solved for The algorithm works by determining a degree bound and then making an ansatz for the solution with undetermined coefficients The command RSolvePolynomial is able to deal with parameters these have to occur linearly in the inhomogeneous part Call the parameterized version using the option ExtraParameters The following options can be given ExtraParameters specifies some extra parameters for which the equation has to be solved EXAMPLES n13 RSolvePolynomial n 3 1 f n 1 3 n 3 n 2 n f n 2 2n 6 f n Out 313 rbl n IH In 314 eqn 1 k f k 1 k n f 1 k 1 k 1 k n 0 1 k 1 n z 1 oupi k n 1 f k 1 k 1 f k k 1D k n Unit k 1 n D a 1 inzis RSolvePolynomial eqn f k out 315j I agi RSolvePolynomial eqn f k ExtraParameters 2x 0 x 1 ou ig f k C 1 k z 0 2C 1 x 1 OD SEE ALSO DSolvePolynomial DSolveRational QSolvePolynomial QSolveRational RSolveRational
43. applied to an inhomogeneous recurrence e g fn4i q q 1 fn q 1 0 niss rec ToOrePolynomial QS qn q n 1 b QS qn qn q 1 5 OreAlgebra GS an q n b Out 85 535 46 qb b Sg b 1 qn neg DF initeQSubstitute rec q h ModuleBasis 2 outes Sanab b Sina Sina d an qd Sana 1 4 La q The result again is to be interpreted as an inhomogeneous recurrence fnas Lad 2 9 fra la 2 Pla P 1 SEE ALSO DFiniteSubstitute QS 25 DFiniteRE2DE DFiniteRE2DE ann nm nab n z4 computes differential equations in x1 q of a generating function whose coefficients satisfy the recurrences in n1 na that are contained in the annihilating ideal ann MORE INFORMATION ann is an annihilating ideal given as a list of OrePolynomial expressions that form a Gr bner basis It is assumed that the operators S Sng are part of the Ore algebra in which ann is represented In the output algebra these shift operators are replaced by the differential operators D Dz The differential equations for the generating function are obtained by relatively simple rewrite rules In general they are inhomogeneous and hence in the next step they are homogenized Finally a Gr bner basis of the resulting operators is computed and returned This procedure does not always deliver the maximal annihi
44. ar and therefore it is recommended to switch to togethered coefficient representation in advance The following options can be given Extended False setting this option to True computes also the cofactors of the reduction The output then is of the form r f c1 09 such that r is the reductum of opoly and opoly can be written as f r c191 292 4 ModuleBasis when computing in a module give here a list of natural numbers indicating where the position variables are located among the generators of the Ore algebra Modulus 0 give a prime number here for modular computations NormalizeCoefficients False whether the content of intermediate and final results shall be removed OrePolynomialSubstitute If a list of rules a ao b gt bo is given then the reduction is com puted with these substitutions note that it takes care of noncommutativity as the substitutions are performed at a point where the noncommuting na ture of these variables is not relevant any more ReduceTail True whether also the tail i e the terms that come behind the first non reducible term should be reduced 70 EXAMPLES npr gb Annihilator LaguerreL n x Der z S n owprz z D n 1 n x 1 n 2 82 2n x 3 S n 1 npr3 OreReduce S n Der x 3 gb n 2n m 2 1 1 n 2n m 2 1 Out 273 rey ar To Ti TU E Sn 2 m oam zs ca
45. ared to first evaluate and then computing an annihilating ideal see below for an example on this issue Similarly if expr contains Sum or Integrate then not Mathematica is asked to simplify the expression but CreativeTelescoping is executed automatically on the summand resp inte grand For evaluating the delta part Mathematica s FullSimplify and Limit are used if they fail or if you don t trust them you can use the option Inho mogeneous in order to obtain inhomogeneous recurrences resp differential equations where the critical components of the delta part have been wrapped with Hold Often the problem is that the evaluation of the delta part requires additional assumptions they can be given with the option Assumptions The following options can be given Assumptions gt Assumptions In cases where CreativeTelescoping is called internally these assumptions are passed and used for simplifying the inhomogeneous parts Head None By default the annihilating operators are returned as OrePolynomial ex pressions if some symbol other than None is given e g Head gt f then the output is given as relations of the specified function Inhomogeneous False Applies only if expr is a sum or an integral In order to present the result as operators the relations found by creative telescoping are homogenized by default If this option is set to True then two lists are returned one contain ing the homogeneous parts
46. as a list of OrePolynomial expressions If the ideal is not zero dimensional the symbol Infinity is returned If exponent vectors are given then also a list of exponent vectors or Infinity is returned If the input consists of OrePolynomial expressions the output is sorted ac cording to the monomial order in which gb is given If the input consists of exponent vectors the output is not sorted EXAMPLES wegl gb Annihilator StruveH n zl owpss 2 D 2nz x S 2nzD n n 4 z xS D n 1 S x 2nz 3x S 4n 10n a 6 5 2 D 3ng 3z nsso UnderTheStaircase gb oupso 1 Dr Sn Insso UnderTheStaircase Take gb 2 Out 360 OO mae UnderTheStaircase ToOrePolynomial 1 OreAlgebra Der x S y Sr n363 Under TheStaircase ToOrePolynomial S a S b S c 2 ow 62 1 Se S5 B Ger Sa Ses Sa Shy Sa DSe aeai Under TheStaircase 2 0 0 0 2 0 0 0 2 aube 0 0 0 0 0 1 0 1 OF 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 SEE ALSO Annihilator Annihilator Dimension OreGroebnerBasis LeadingExponent LeadingPowerProduct LeadingTerm Support 92 References 1 Sergej A Abramov Rational solutions of linear differential and difference 10 1 12 equations with polynomial coefficients USSR Computational Mathematics and Mathematical Physics 29 6 7 12 198
47. asis of the resulting oper ators is computed and returned Alternatively it is conceivable to use Cre ativeTelescoping or Takayama on Cauchy s integral formula that extracts the coefficients of the generating function see the examples below EXAMPLES n54 DFiniteDE2RE Annihilator Log 1 x Der x n oupa n 1 S n n55 DFiniteDE2R E Annihilator E x y Der z Der y x y m ail out ssj n 1 S L m 1 Sm 1 isel DFiniteDE2RE Annihilator BesselJ n x m ouso 5 m n 1 Sm m Am n 482 1 ish CreativeTelescoping BesselJ n 2 z rn 1 Der z S n S m one IS m n4 285 m 4m n 482 Das RH nssj ann Annihilator BesselJ n z z m 1 Der z S n S m n59 Takayama ann i t ouso m n 1 Sm m Am n 4 82 1 SEE ALSO CreativeTelescoping DFiniteRE2DE Takayama 19 DFiniteOreAction DFiniteOreAction ann opoly computes an annihilating ideal for the function that is obtained when the operator opoly is applied to the function described by ann using the closure property application of an operator MORE INFORMATION The closure property application of an operator reads as follows Let f be a function that is 0 finite with respect to the operators 01 04 then Pe f is O finite as well where P is an operator in the Ore algeb
48. at occur in expr p 61 OrePlus opoly opoly2 adds the Ore polynomials opoly and opoly2 p 62 OrePolynomial data alg order is the internal representation of Ore polyno mials p 63 OrePolynomialDegree opoly gives the total degree of the Ore polynomial opoly p 65 OrePolynomialList Coefficients opoly gives a list containing all nonzero co efficients of the Ore polynomial opoly p 66 OrePolynomialSubstitute opoly rules applies the substitutions rules to the Ore polynomial opoly p 67 OrePolynomialZeroQ opoly tests whether an Ore polynomial is zero p 68 OrePower opoly n gives the n th power of the Ore polynomial opoly p 69 OreReduce opoly Jo os D reduces the Ore polynomial opoly modulo the set of Ore polynomials 91 g2 p 70 OreSigma op defines the endomorphism c for the Ore operator op p 72 OreTimes opoly1 opoly multiplies the Ore polynomials opoly and opoly respecting their noncommutative nature p 73 Printlevel n activates and controls verbose mode see p 74 QS 2 qn represents the q shift operator on x p 75 QSolvePolynomial leqn fle q determines whether the linear q shift equation eqn in f a with polynomial coefficients has polynomial solutions and in the affirmative case computes them p 76 QSolveRational eqn f x a determines whether the linear q shift equation eqn in f x with polynomial coefficients has rational solutions
49. ativeTelescoping ann deltas finds creative telescoping relations in the annihilating ideal ann MORE INFORMATION The first argument is either an annihilating ideal i e a Gr bner basis of such an ideal or any mathematical expression In the latter case Annihilator is inter nally called with expr The second argument deltas indicates which summations and integrations have to be performed this means S a 1 resp QS qa da 1 for q summation w r t the variable a and Der a for integration w r t a If expr is given then the third argument ops specifies the surviving Ore opera tors i e the operators that occur in the principal part as well as in the delta part For a more detailed explanation of the method of creative telescoping see CreativeTelescoping p 14 For a more detailed description of how the ansatz used in this command is constructed see the paper 10 The output consists of two lists L1 L3 where L contains all the principle parts such that they constitute a Gr bner basis and L contains the corre sponding delta parts In particular Li and Lo have the same length and the i th element of L is the list of delta parts corresponding to the given deltas for the i th principal part in L4 The following options can be given Mode Automatic This command can be used in different ways Automatic everything is done automatically i e the minimal ansatz is determined in
50. be an OrePolynomial expression or a plain poly nomial in the existing Ore operators also a list of the previously mentioned is admissible How the occurring Ore operators act on expr is defined by the com mand OreAction The second argument can be any Mathematica expression to which the necessary actions can be applied If opoly contains q shift opera tors of the form QS z q then all occurrences of the dummy variables x are replaced by q EXAMPLES npn opoly ToOrePolynomial Der x 2 1 nps ApplyOreOperator opoly Sin oupsj U npg Apply OreOperator S n n 1 n oups n 1 n n 1 npo ApplyOreOperator qn a n outf30j G Qn ai Apply OreOperator ToOrePolynomial q OreAlgebra QS qn q 4 2n out 31j d SEE ALSO Delta Der Euler OreAction QS ToOrePolynomial S 11 ChangeMonomialOrder ChangeMonomialOrder opoly ord changes the monomial order of the Ore polynomial opoly to ord MORE INFORMATION The input opoly must be an OrePolynomial expression or a list of such The terms of opoly are reordered according to the new monomial order ord and the OrePolynomial expression s carrying the new order is returned The following monomial orders are supported See the description of Ore GroebnerBasis p 57 for more details Lexicographic ReverseLexicographic DegreeLexicographic DegreeReverseLexicograp
51. bose mode using Printlevel Trying a higher degree results in a shorter recurrence inga5j ann Annihilator q i 7 2 2 QPochhammer q q n i j QPochhammer q q i QPochhammer q q j QS qi a i QS qj 4 3 QS qn q n oupas qi qj AN Sn a qi qj qaj 1 Si a d qi aj q ai qj qn qai 1 8 qai qj qai gj qu in346 Takayama ann 1 j oupas 1 aan S ay m grad Ne SE E REUS q an m Lu tat T2q qn q qn d qn qqn qqn4 d 1 pe T tg SC 4 3 6 qqu q qn g qng 0 0 0 Rng T T inga7j Takayama ann qi qj Start Degree 7 Out 3a7 I aqn US q qn don don don q qn g an q q 1 Saa don go d an d 4 a Sng L N Oo I In the last example the input is not holonomic and hence Takayama would run for ever if not given the option MaxDegree nbs Takayama Annihilator 1 k n S k S n k MaxDegree 20 ourzasj Failed SEE ALSO Annihilator CreativeTelescoping OreGroebnerBasis Printlevel 89 ToOrePolynomial ToOrePolynomial expr converts expr to an OrePolynomial expression ToOrePolynomial lexpr al g converts expr to an OrePolynomial expression in the Ore algebra alg ToOrePolynomial leqn flu Uk converts the equation of f into operator notation MORE INFORMATION The input expr must be a polynomial in
52. der opoly ord changes the monomial order of the Ore polynomial opoly to ord p 12 ChangeOreAlgebra opoly alg translates the Ore polynomial opoly into the new Ore algebra alg p 13 partially supported by the Austrian Science Fund FWF P20162 N18 and by the US National Science Foundation NSF DMS 0070567 CreativeTelescoping ezpr delta ops performs Chyzak s algorithm to find creative telescoping relations for expr p 14 Delta n represents the forward difference delta operator w r t n p 17 Der 2 represents the operator partial derivative w r t x p 18 DFiniteDE2RE ann 21 s a m na computes recurrences in the variables n1 nq for the coefficients of a power series that is solution to the given differential equations in ann p 19 DFiniteOreAction ann opoly executes the closure property application of an operator for the 0 finite function described by ann p 20 DFinitePlus anni ann executes the closure property sum for the O finite functions described by ann and ann p 22 DFiniteQSubstitute ann q m kj computes an annihilating ideal of q difference equations for the result of the substitution d e 7 mg p 24 DFiniteRE2DE ann n Ma Tiye ta gives differential equations in 1 q of a generating function whose coefficients are described by the given recurrences in ann p 26 DFiniteSubstitute ann subs executes the closure
53. e commands Expand Factor and Together can be applied to an OrePolynomial expression However they affect only the coefficients In particular Factor p does not perform any operator factorization It just causes that the coefficients will be given in factored form Note that this operation also changes the OreAlgebraObject of p in order to remember that the coefficients are factored In all subsequent operations this property then will persist Some standard polynomial operations can be performed on OrePolynomial ex pressions like on standard polynomials They include Coefficient Exponent and Variables The command PolynomialMod is applied to the coefficients 63 only Finally Normal p converts an Ore polynomial to a standard commuta tive polynomial EXAMPLES nps alg OreAlgebra S n Der z oupsi Kin x Sn 4 0 Ds 1 Dz nps pl ToOrePolynomial S n n alg Oup32 S n ip33 p2 ToOrePolynomial z Der x S n 1 alg oupas Sn xD 1 In 234 p1 p2 out 234 25 T D n 1 In 235 p1 p2 oupss T Dr n 1 In 236 P p2 Out 236 S xS D n 1 5 nz D n In 237 p2 p1 Out 237 S2 Sn Der 4 nS nzD n npssj First 96 o ps 1 12 0 o 5 13 n 1 OF 02 0 1 70 10 055 ipa n 1 p1 3 owpss n 1 52 3n 6n 3 8 3n 6n 4n UR nf n np40j Factor ompa n 1 8 3 n 1
54. e elimi nated i e they are lexicographically greater than the rest In the two blocks the variables are ordered by DegreeLexicographic WeightedLexicographic w weighted lexicographic ordering the total degrees weighted by the weight vector w are compared first and if they are equal the two terms are compared lexicographically WeightedLexicographic w order similar as before but now the two terms are compare by order in case that their weighted total degrees are the same MatrixOrder m matrix ordering the exponent vectors are multiplied by the matrix m and the results are compared lexicographically You can define your own monomial ordering by giving a pure function that takes as input two exponent vectors and yields True or False Extended False setting this option to True computes also the cofactors of the Gr bner basis that allow to write each of its elements as a linear combination of the P In this case a list of length 2 is returned its first element being the Gr bner basis and its second element being the matrix of cofactors Incomplete False setting this option to True causes that the computation is interrupted as soon as an element free of the elimination variables is found works only in connection with EliminationOrder ModuleBasis when computing in a module give here a list of natural numbers indicating where the position variables are located among the generators of the Ore algebr
55. ecifies the Ore algebra in which the output should be given None means that the Ore algebra of the input is taken MonomialOrder None specifies the monomial order in which the output should be given None means that the monomial order of the input is taken EXAMPLES mp annl Annihilator Log Der z 27 owpu xD De iaai ann2 DFiniteSubstitute ann1 x y 2 z 1 3 Algebra OreAlgebra Der y Der z o p y Dy D 62D 6D 362 D yD 362D y D 27y z 272 DI 54992 10827 D2 y D 69 542 Dr went Simplify ApplyOreOperator ann2 Log y 2 z 1 3 ourf3 0 0 0 Continuous and discrete substitutions can be performed in one single step mp4 annl Annihilator LegendreP n zl S n Der ax owpa n 1 1 2 D n x x 1 D2 2x D n n hps ann2 DFiniteSubstitute ann1 n 3n k x Sqrt y 2 1 Algebra OreAlgebra S k S n Der y ag FullSimplify ApplyOreOperator ann2 LegendreP 3n k Sqrt y 1 outos 0 0 0 0 0 0 A recurrence for n 2 iaz DFiniteSubstitute ToOrePolynomial S n n 13 n gt n 2 Out 97 25 n 2 The closure property substitution includes computing diagonals hpg DFiniteSubstitute Annihilator Binomial 2n k S k S n k gt n Algebra OreAlgebra S n oups n 1 S 4n 2
56. ed 24 EXAMPLES We study a very simple example the sequence Ld q n ny7 ann Annihilator QPochhammer g q n QS an q n oupz San a 1 d dn L The command DFiniteQSubstitute is now employed to study the sequence q q n which corresponds to the substitution q e27 q ral ann2 DFiniteQSubstitute ann q 2 outral Sing 1 q Sing q dont in7 ApplyOreOperator ann2 QPochhammer gq q n n gt 4 FullSimplify oarat 0 nso DFiniteQSubstitute ann q 2 Return Backsubstitution Out 80 Sina F 1 q San a q do ent ann2 OrePolynomialSubstitute q q ouis True Next we twist by a third root of unity Note that in the result all the powers of qn are divisible by 3 ng ann Annihilator QBinomial 2n n q QS qn q n oct 1 GN Raa 1 gan qan q qn ni3 ann3 E Jo 3 Return gt rer Out 83 d on 24 an 3q7qu R I qn 2g qu Rn qn cn qn BEN um 20 an rie an 20 ol q Kat VE oan me qu 2q qn SUR ant 70 an 3q mm Aal qu Zo qn 4 29 qn 4g qn q qn 29 qn Ae om M qn 3g qn 4q 29 2 8 3 ap qn Ag qn gon 2 qn 24 qu 24 qn 29 qn 497 2q 2q misal Union Cases ann3 qn Infinity 3 6 9 12 15 18 Out 84 qn qn qn qn qn qn Now we demonstrate how this procedure can be
57. ent are the names of these functions and in the third argument the variables on which the f depend SolveCoupledSystem uses OreGroebnerBasis to uncouple the given system which corresponds to Gaussian elimination The advantage compared to SolveOreSys is that more general types of systems can be dealt with i e there are no restrictions concerning the order and even mixed difference and differential systems can be addressed Additionally this command is also more reliable than SolveOreSys Note that SolveCoupledSystem adresses only linear systems if the input is not linear which is not checked the result most probably will be wrong The following options can be given ExtraParameters specifies some extra parameters for which the equations have to be solved these parameters are allowed to occur in eqns only linearly and only in the inhomogenous parts Return solution specifies what result should be returned by default this is the solution of the eqns Return uncoupled returns the uncoupled system without solving EXAMPLES in327 SolveCoupledSystem n 1 f z x nz g z 272 1 g z 0 a na f z n 1 g z z 2 1 f r 1 2 2 f g x Out 327 Ja ES ale gt Sie azgi SolveCoupledSystem n 2 a 1 n n 2 b n 2n 3 b 1 n 0 n 2 a n n 1 b 1 n n 2 0 a b n outt32e a n C 1 n
58. er r Expand Function Plus Slot 1 Slot 2 Function Expand Times Slot 1 Slot 2 None n85 ToOrePolynomial z x Der 1 alg ou ssj 7D 1 hagi ChangeOreAlgebra OreAlgebra Der z l ou e9g Dx inf190 OreAlgebra Out 190 K a Dz 1 Di npo OreAlgebra S k Delta n Der Euler z QS qn q n ouf191 K k n qn x z Sk Sk 0 A Sn An De 1 Ds 02 1 02 Sqn ai 94 4 0 m93 OreAlgebra n S n Der a Out 192 K x n 95 Sn 0 Dz 1 Di SEE ALSO OreAction OreAlgebraGenerators OreAlgebraOperators OreAlgebraPolynomialVariables OreDelta OreOperatorQ OreOperators OrePolynomial OreSigma ToOrePolynomial 52 OreAlgebraGenerators OreAlgebraGenerators al 9 gives the list of generators of the Ore algebra alg MORE INFORMATION The generators are in the same order as they have been given when creating alg using OreAlgebra Taking into account the internal structure how an Ore algebra is represented OreAlgebraGenerators is the same as First EXAMPLES np193j alg OreAlgebra p Der x n S n Delta m x ouf193 K m p n z Ds 1 Dz Sn Sn 0 Am Sm Am np194j OreAlgebraGenerators alg ouha p Der x n S n Delta m x nposj First alg out195 p Der x n S n Delta m x SEE ALSO OreAlgebra OreAlgebraOperators OreAlgebraPolynomialVariables 53 OreAlgebraOperator
59. f generators no division by those is allowed The endomorphisms o and 6 are not given explicitely when creating an Ore algebra They have to be attached to the symbols 0 before using the com mands OreSigma OreDelta and OreAction For standard applications this is not necessary since the common Ore operators are already predefined The order in which the generators o 92 are given is used when an Ore poly nomial is represented in this algebra all power products are written according to this order For example when dealing with the Weyl algebra you can choose whether x appears always on the left and D on the right the standard way or the other way round The coefficients in the general form as stated above they are elements in K z1 24 are always written on the left When working in a module over some Ore algebra use position variables to represent the element Tj Tm T O write it either as pi Ti PmIm with new indeterminates p1 Pm position variables or as pT p T p T with a single position variable p The position variables have to be added to the generators of O and their positions among the generators have to be given with the option ModuleBasis to the relevant commands that can deal with modules OreGroebnerBasis FGLM FindRelation etc An Ore algebra is displayed using the standard mathematical notation for such algebras but internally it is represented as a Mathe
60. gExponent LeadingPowerProduct NormalizeCoefficients OrePolynomial OrePolynomialListCoefficients ToOrePolynomial Support 48 NormalizeCoefficients NormalizeCoefficients opoly removes the content of the Ore polynomial opoly MORE INFORMATION The input opoly has to be an OrePolynomial expression The output is a multiple of the input such that all denominators are cleared and the content is 1 When the option Extended is used then the output is a list containing the content and the normalized Ore polynomial The following options can be given Denominator True setting this option to False deactivates the computation of a common de nominator can be used for better performance when it is known in advance that the input has polynomial coefficients Extended False when you also need the content that has been removed from opoly then set this option to True the output then is of the form c op where c is the content given as a standard Mathematica expression and p is the normalized Ore polynomial given as a OrePolynomial expression Modulus 0 if a prime number is given with this option then all polynomial operations like PolynomialGCD or Together are performed modulo this prime EXAMPLES np72 opoly Together ToOrePolynomial n n 1 S n 2 n 1 n 1 S n n 2 1 n 1 n l npr3 NormalizeCoefficients opoly outi7gj n n S2
61. gorithm 6 where the uncoupling is done by Gaussian elimination using the OreGroebnerBasis command usu ally the fastest and most reliable option AbramovZima Gauss Zuercher IncompleteZuercher Chyzak s al gorithm but uses Stefan Gerhold s implementation 7 of different un coupling algorithms his package OreSys has to be loaded in advance Barkatou Chyzak s algorithm but uses Barkatou s algorithms 3 4 to solve the system directly without uncoupling since we implemented only some cases of these algorithms this option does not work in most cases Hermite Creative telescoping algorithm for hyperexponential func tions 5 based on Hermite reduction Return Automatic the value of this option is passed to the command HermiteTelescoping when Method Hermite is used Support specify the support of the principal part P 15 EXAMPLES ail CreativeTelescoping Binomial n k S k 1 S n om 8 2 rr l az CreativeTelescoping ChebyshevT n 1 x 2y Sqrt 1 x 2 Der z S n Der y Out 42 os 2n Sa 2ny Any y 2y D 2n y 2n ny 2n Y 2y Dj y 2 D n 4 ERE 2 EH The following allows us to write down the indefinite integral of the Hermite polynomials namely f H x dx Hn 1 x 2 n 1 n3 CreativeTelescoping HermiteH n x Der x S n ouh a Jens The next example is famo
62. gument subs has to be a list of Rule expressions DFiniteSubstitute executes the closure properties al gebraic substitution for continuous variables and rational linear substitution for discrete variables An algebraic substitution is given in the form x expr where x is a con tinuous variable i e Der x is part of the algebra of ann and where expr is algebraic in 21 22 the continuous variables of the output specified by the option Algebra i e there exists a polynomial p a 21 z2 such that p expr 21 22 0 Note that the variable z must not appear on the right hand side of the substitution rule substitutions like x gt vz or x x y are not admissible Instead new variables have to be introduced such as x gt vz orrz gt yt Z A discrete variable n i e S n is part of the algebra of ann can be replaced via n expr where the expression expr is rational linear in the discrete variables k ko of the output expr c rik raka where r Q and c is an arbitrary constant Note that for discrete substitutions the variable to be replaced is allowed to appear on the right hand side i e substitutions like n 2n or n 2 k n 2 are valid In cases where the set of variables is not changed the option Algebra needs not to be given DFiniteSubstitute is called by Annihilator whenever a substitution has to be performed The following options can be given Algebra None sp
63. he operator partial derivative w r t x MORE INFORMATION When this operator occurs in an OrePolynomial object it is displayed as D The symbol Der receives its meaning from the definitions of OreSigma Ore Delta and OreAction EXAMPLES magi OreDelta Der z out 49 Or HI amp nso Apply OreOperator Der z 2 f z ouso f 2 The symbol Der itself does not do anything In order to perform noncommu tative arithmetic it first has to be embedded into an OrePolynomial object In b1 Dere kg oupij Der s x n53 ToOrePolynomial Der a out 52 Dy In 53 ek 0 outs3j 7D 1 SEE ALSO ApplyOreOperator Delta Euler OreAction OreDelta OreSigma QS S ToOrePolynomial 18 DFiniteDE2RE DFiniteDE2RE ann Jr 2a Im na computes recurrences in n1 Na for the coefficients of a power series that is solution to the given differential equations in 71 2q in ann MORE INFORMATION The input ann is an annihilating ideal given as a list of OrePolynomial expres sions that form a Gr bner basis It is assumed that the operators D Dz are part of the Ore algebra in which ann is represented In the output alge bra these differential operators are replaced by the shift operators S Sna The recurrences for the coefficients of the generating function are obtained by relatively simple rewrite rules Finally a Gr bner b
64. hic EliminationOrder n WeightedLexicographic w WeightedOrder w order MatrixOrder EXAMPLES azi opoly ToOrePolynomial Sum S a i S b 3 2 0 2 7 0 3 OreAlgebra S a S b MonomialOrder DegreeLexicographic ompa Se Sp SS S S Ses Say Sy Se SoS S Sa ESI ns3 ChangeMonomialOrder opoly Lexicographic onp E SESS SoS S Susy Sa Sih Sa t SP TS 541 aal ChangeMonomialOrder opoly WeightedOrder 2 1 Lexicographic onp Sp RR LRR LRR R LRR SoS ES Sa S ESI SEE ALSO ChangeOreAlgebra FGLM OrePolynomial ToOrePolynomial 12 ChangeOreAlgebra ChangeOreAlgebra opoly al 9 transforms the Ore polynomial opoly into the Ore algebra alg MORE INFORMATION To each OrePolynomial expression the algebra in which it is represented is attached The command ChangeOreAlgebra can be used to change this rep resentation if possible Note that the transformed Ore polynomial is returned and not the original opoly is changed Changes can concern the order in which the generators of the algebra are given as well as the set of generators itself The command may fail for several reasons e g if the input is not a polynomial in the new set of generators or if the input involves some Ore operators that are not any more contained in alg in such cases Failed is returned The following options can be given Monomial
65. hich the modular computations are done EXAMPLES See FindRelation for more information about the following example Note also that computing the full relation there takes 80 seconds whereas finding the support only is much faster 41 np36 ann Annihilator LegendreP 2k 1 x x 2 Der z S k hai Timing FindSupport ann Eliminate a oup37 3 10019 Der z S k Der a S k 7 Der z S k SIVI Der E Der x 2 Si Dere S Ky S k Det a Delai S k Der z Der x S k S k Der x S k SK S k SIK Der x zl S k Herle a NO ie Dera gi Der z ai Der 2 S k Der z 9 S k Deal S k E Der z S k Ders S k x Der a S k Der a S k For more examples see FindRelation its use is very similar to FindSupport SEE ALSO Annihilator FindCreativeTelescoping FindRelation OreGroebnerBasis Printlevel Support 42 GBEqual GBEqual 901 gb compares two Gr bner bases gb and gb gt whether they are the same MORE INFORMATION The two Gr bner bases gb and gby have to be given with respect to the same monomial order and they have to be completely reduced Both are lists of OrePolynomial expressions GBEqual does not perform any advanced math ematics it just goes through the two lists and checks whether either the sum or the difference of the corresponding elements gives 0 Hence it gives True if the corresponding Gr bner ba
66. hilated by a first order recurrence But the output of the closure property DF initePlus is a recurrence that annihilates any linear combination c n 4 1 canl and hence is of order 2 Note that in this case also DFiniteOreAction could be used to obtain the first order recurrence n3 DFinitePlus OG ToOrePolynomial S n n 2 S n n 1 outraj S7 2n 4 S n 3n 2 al Annihilator n 1 n S n owpa S7 2n 4 S n 3n 2 n sj Annihilator n n S n pwal nS 2n 2n 1 rel DFiniteOreAction Annihilator n S n S n 1 owrg nS n 2n 1 SEE ALSO Annihilator DFiniteOreAction DFiniteSubstitute DFiniteTimes 23 DFiniteQSubstitute DFiniteQSubstitute ann 10 m computes an annihilating ideal of q difference equations for the result of the substitution q gt e mq DFiniteQSubstitute ann q m kj computes an annihilating ideal of q difference equations for the result of the substitution q gt e2 7 mgl k DFiniteQSubstitute ann Gn mila das Ma ka does the same for several such substitutions MORE INFORMATION ann is an annihilating ideal given as a list of OrePolynomial expressions that form a Gr bner basis Typically ann consists of q difference equations involving the Ore operator QS The command DFiniteQSubstitute first finds some elements
67. i RSolveRational eqn f k out 319j asni RSolveRational eqn f k ExtraParameters x 0 1 vu lan gt LL 0 gt 2cti aft ol SEE ALSO DSolvePolynomial DSolveRational QSolvePolynomial QSolveRational RSolvePolynomial 80 S S n represents the forward shift operator with respect to n MORE INFORMATION When this operator occurs in an OrePolynomial object it is displayed as Sn The symbol S receives its meaning from the definitions of OreSigma Ore Delta and OreAction EXAMPLES np2 OreSigma S n oupa Z1 n n 41 amp np2 OreDelta S n our 322 U amp nsz23j ApplyOreOperator S n 3 f n ous23j f n 3 The symbol S itself does not do anything In order to perform noncommutative arithmetic it first has to be embedded into an OrePolynomial object In324j S n n 2 oupzaj S n za n azgi ToOrePolynomial S n 0ut 325 Sn Inf326 Yo 1 72 Out 326 n 2n 1 S SEE ALSO ApplyOreOperator Delta Der Euler OreAction OreDelta OreSigma QS ToOrePolynomial 81 SolveCoupledSystem SolveCoupledSystem eqns Ip fr v1 och computes all rational solutions of a coupled system of linear difference and differential equations MORE INFORMATION The first argument eqns is a list of linear equations in the functions fi fk their shifts and or their derivatives the second argum
68. imination a greedy strategy that is designed for elimination problems it prefers pairs where the total degree of the variables to be eliminated is minimal in the lcm of the leading monomials pairsize pairs of small size are treated first we take as the size of a pair the product of the ByteCounts of the two polynomials 57 MonomialOrder None specifies the monomial order None means that the same monomial order is taken in which the input is given If the input is not given as OrePoly nomial expressions then DegreeLexicographic is taken per default The following monomial orders can be chosen DegreeLexicographic total degree lexicographic order graded lexi cographic ordering The terms of higher total degree come first two terms of the same degree are ordered lexicographically see below DegreeReverseLexicographic total degree reverse lexicographic order ing Lexicographic the terms with the highest power of the first variable come first for two terms with the same power of the first variable the power of the second variable is compared and so on ReverseLexicographic reverse lexicographic ordering The term with the higher power of the last variable comes last for terms with the same power of the last variable the exponent on the next to last variable is compared and so on Note Under this ordering the monomials are not well ordered EliminationOrder n a block order s t the first n variables ar
69. ion RSolveRational got a recurrence of order 1 RSolvePolynomial degree bound 0 Solved scalar equation CreativeTelescoping Trying d 1 ansatz eta 0 1 eta 1 S n 1 S k phi 1 k 1 LocalOreReduce Reducing 0 1 Start to solve scalar equation RSolveRational got a recurrence of order 1 RSolvePolynomial degree bound 1 Solved scalar equation oupos S 2 Out 294 n nx 74 QS QS qn represents the q shift operator on the variable x MORE INFORMATION The q shift on x is defined to be qx Often in practice the variable x is in fact a power of q e g z q and then the q shift of x corresponds to a shift in n That s why the Ore operator QS takes two arguments namely the variable on which the g shift acts and the power of q to which this variable corresponds Since for polynomial arithmetic it is problematic to deal with indeterminates like q such occurrences will always be replaced by the variable x e g when cre ating an OrePolynomial expression When it is part of an OrePolynomial the q shift operator is displayed as S When applying it to some expression the variable x is again replaced by q The operator QS z q acts on x via z gt qx as well as on n via n gt n 1 This behaviour is important when you deal with expressions like QBinomial n k al where the variable x or the power q do not appear explicitely EXAMPLES inp
70. ion Modulus See the example below 37 EXAMPLES hast Find CreativeTelescoping Binomial n k S k 1 S n ane sn fot When using uncoupling to compute creative telescoping relations as it is done in CreativeTelescoping then the following example is very hard to solve but using FindCreativeTelescoping it is solved in a few seconds np24j ann Annihilator x BesselJ 1 ax Bessell 1 ax BesselY 0 x BesselK 0 x Der Der a np25j iming First FindCreativeTelescoping ann Der oupps 10 7167 a Da 2 Compute a creative telescoping relation for the Andrews Paule double sum In 126 FindCreativeTelescoping Binomialf i j 2 2 Binomial 4n 2i 2j 2n 2i S 1 S j 1 S n 2i j i n i 2ij 3ijn 2ij 3in out 126j 4 1 z S G 1 i j 2n 2i j ij 3ijn 2ij j n j 3jn yl 1 6 j 2n H The following example illustrates the use of the options FindSupport and Modular Assume ann is the annihilating ideal for some function f n k that is to be summed over k and the direct computation of the creative telescoping relation is too expensive e g requires more memory than available Then the following commands can be used to generate the data that is necessary for reconstructing the solution from homomorphic images In the example 80 interpolation points and 10 prime numbers are u
71. lating ideal for the generating function and therefore it is conceivable to use CreativeTelescoping or Takayama for this task see the example below EXAMPLES n7 DFiniteRE2DE ToOrePolynomial S n 19 n x oui z 1 D 1 The following example shows how much DFiniteRE2DE can overshoot With CreativeTelescoping we find the maximal annihilating ideal for the generat ing function west DFiniteRE2DE Annihilator LegendreP n x S n Dere n y owgs z y y D Dy xy y D a 1 D Bay 1 D 2 1 2 D2 y D 2xD 2yDy 2zy y y Dj Tay 6y y Dy 2 HUD 2 y Ty 1 D y Insg Creative Telescoping LegendreP n z y n S n 1 Der Der y Outfs9 rz t y c 1 D y 2 en y 1 D y le 1 x 1 D 4 E xy 1 D 2i npo Takayama A nnihilator LegendreP n zum S n Der z Der y n owpo z 1 zy 1 D z 2xy y 1 D 3y 3x D 1 SEE ALSO CreativeTelescoping DFiniteDE2RE Takayama 26 DFiniteSubstitute DFiniteSubstitute ann subs computes an annihilating ideal for the function that is obtained by perform ing the substitutions subs on the function described by ann MORE INFORMATION The input ann is an annihilating ideal given as a list of OrePolynomial ex pressions that form a Gr bner basis The second ar
72. left ideal that is generated by the Ore polynomials P Pr OreGroebnerBasis Up sje Pee alg translates P P into the Ore algebra alg and then computes a left Gr bner basis MORE INFORMATION The input polynomials P4 P have to be given as OrePolynomial expres sions If they are not in this format or if they belong to different Ore algebras then the second argument has to be given to specify in which algebra the com putation should take place OreGroebnerBasis executes Buchberger s algorithm and it makes use of the chain criterion Buchberger s second criterion The product criterion Buch berger s first criterion cannot be used in noncommutative polynomial rings Per default the output is a completely auto reduced Gr bner basis all its ele ments are normalized to have content 1 and they are ordered by their leading monomials according to the monomial order The following options can be given Method sugar specifies the pair selection strategy in Buchberger s algorithm The following methods can be chosen sugar the sugar strategy has been proposed by Giovini Mora Niesi Robbiano and Traverso in 1991 and is one of the most popular pair selection strategies It mimics the Gr bner basis computation in an homogeneous ideal normal the normal strategy has been proposed by Bruno Buchberger himself it takes the pair with the smallest lcm of the leading monomials el
73. lgebra these can be influenced by the option Support ModuleBasis when ann is a Grobner basis over a module give here a list of natural num bers indicating the location of the position variables among the generators of the Ore algebra Pattern gt include only monomials into the ansatz whose exponent vectors match the given pattern Support Automatic in the default setting the total degree of the support is increased until some thing is found alternatively you can give a list of power products of the generators of the Ore algebra that are used in the ansatz EXAMPLES Find contiguous relations of the 5 F1 hypergeometric function m25 Annihilator Hypergeometric2F1 a b c x S a S b S c Der owp ab ac bc S c cx Dy ac be ba xD b aSa xD a a x D ax bx c x D ab np30 FindRelation Support 11 S a S c omoj acx ac S abr acz ber c z 8 ac ber c 2 39 For solving Strang s integral DU Oa 2 2 dr with Zeilberger s slow algo rithm we would like to eliminate z from the annihilating ideal of the integrand of course this is not the best way to solve this integral see CreativeTele scoping p 14 The output is of considerable size that s why we display only its support to illustrate why FindRelation takes some time here np3 ann A
74. lynomial Der z x OreAlgebra Der z x 90 out 352 Dr x inj353 ChangeOreAlgebra OreAlgebra 0ut 353 7D 1 Inss4j ToOrePolynomial S n sa 1 n 3 1 Out 354 min Insssj ToOrePolynomial S n xx 1 n 3 OreAlgebra n S n ToOrePolynomial nopoly The input is not a polynomial w r t generators of the algebra ou sss Failed mapel ToOrePolynomial Der 1 2 x f x 2 3 ougsoj f a 3 D2 21 a 622 D f a 62 nps7 ToOrePolynomial S n n OrePower exp Unvalid exponent n ToOrePolynomial badalg The input cannot be represented in the corresponding algebra ouian Failed SEE ALSO OreAlgebra OreOperatorQ OreOperators OrePolynomial 91 UnderTheStaircase UnderTheStaircase 90 computes the list of monomials power products that lie under the stairs of the Gr bner basis gb UnderTheStaircase exps computes the list of exponent vectors that lie under the stairs defined by the exponent vectors exps MORE INFORMATION The input gb is a list of OrePolynomial expressions The command Under TheStaircase just looks at their leading power products and assumes that they in fact form a Gr bner basis this property is not checked Alternatively the exponent vectors of the leading power products can be given If the left ideal that is generated by gb is zero dimensional then the power products that lie under its stairs are returned
75. matica expression of the form OreAlgebraObject gens cNormal cPlus cTimes ext Herein gens is a list containing the generators of the algebra The next three arguments are 51 functions that determine the coefficient arithmetic cNormal specifies in which form the coefficients should be kept e g in expanded or factored form in particular cNormal is supposed to yield 0 whenever a coefficient is actually zero recall that expressions like 1 PE sit are not automatically simplified to 0 The functions cPlus and cTimes are used for adding resp multiplying two coefficients Finally ext tells whether to work in an algebraic extension For easily setting the coefficient representation certain upvalues have been defined for Ore algebras e g Together alg changes the functions cNormal cPlus and cTimes to Together Analogously for Expand and Factor The following options can be given Coefficient Normal Expand specifies in which form the coefficients of Ore polynomials in this algebra are represented Coefficient Plus 1 2 amp specifies how to add two coefficients of Ore polynomials in this algebra CoefficientTimes Expand 71 2 amp specifies how to multiply two coefficients in this algebra Extension None give an algebraic extension ezt EXAMPLES npsej alg OreAlgebra z Der z Out 186 K x Dz 1 Di nps7 FullForm alg ou i57 OreAlgebraObject List r D
76. maximal degree with respect to vars for which it is tried to find operators free of vars in the truncated module set it to a finite number to prevent an infinite loop EXAMPLES We start with Moll s quartic integral z o x4 2ar 1 41 mgg ann Annihilator 1 z 2a2 1 m 1 S m Der z Der a ouis 2ax z UD 2ma 22 2ax 2 1 D 4amz 4ax Ama 42 Qax x4 1 Sn 1 aagi Takayama ann Out 339 4m 4 Sm 2aD 4m 3 ingao Takayama ann x Extended True ouso 4m 4 Sm 2aDa 4m 3 Gaz a L Sin HI aan Takayama ann x Extended True Reduce True oufsaj 4m 4 Sn 24D 4m 3 aagi Takayama ann x Saturate True owl 4m 4 Sm Zo 4m 3 4a 4 D2 8am 12a Da 4m 3 With Takayama s algorithm multiple sums and integrals can be done in one stroke e g ent rms H z H x oy L Xx E la vie m 0 n 0 inga3j ann Annihilator HermiteH m zl HermiteH n x r m a n Exp 2x 2 m n S m S n Der z Der r Der s ingaqj Takayama ann m n x Out 344 D 2r D 2s 88 For q summation either the summation variables can be given or their corre sponding q powers In the first example the fourth order recurrence is found at degree 6 can be seen when switching to ver
77. mp mps3 OreAction S2 a OreSigma S2 a nps4 ToOrePolynomial S2 n n 2 Out 184j 92n n niss ApplyOreOperator 96 f n Out 185 n f n f n 2 SEE ALSO ApplyOreOperator OreDelta OreOperatorQ OreOperators OreSigma 50 OreAlgebra OreAlgebra o 925 sl creates an Ore algebra with the generators g1 92 OreAlgebra opoly gives the Ore algebra in which the Ore polynomial opoly is represented OreAlgebra ann gives the Ore algebra in which the annihilating ideal ann is represented MORE INFORMATION In the above specification opoly is assumed to be an OrePolynomial expres sion and ann to be a list of such The elements of ann usually will be represented in the same Ore algebra so this algebra will be returned However if you give a list of OrePolynomial expressions that do not share the same algebra you get back a list of Ore algebras corresponding to the elements of the input ann The command OreAlgebra is also used to create an Ore algebra For this purpose the generators of the desired Ore algebra have to be given as argu ments By the generators of an Ore algebra we understand all variables and Ore operators that occur polynomially An Ore algebra in general has the form K z1 zt ellen 61 104 oa 94 the generators of this Ore al gebra are y1 Yn 01 04 Note that the Ore operators 0 always have to be included into the set o
78. ms ap proach PhD thesis Research Institute for Symbolic Computation RISC Johannes Kepler University Linz Austria 2009 Christoph Koutschan A fast approach to creative telescoping Mathematics in Computer Science 4 2 3 259 266 2010 Nobuki Takayama An algorithm of constructing the integral of a module an infinite dimensional analog of Gr bner basis In Proceedings of the In ternational Symposium on Symbolic and Algebraic Computation ISSAC pages 206 211 New York NY USA 1990 ACM Doron Zeilberger A holonomic systems approach to special functions iden tities Journal of Computational and Applied Mathematics 32 3 321 368 1990 93
79. nator Following Abramov s algorithm 2 first the denominator of the solution is determined Then QSolvePolynomial is called to find the numerator polynomial The command QSolvePolynomial is able to deal with parameters these have to occur linearly in the inhomogeneous part Call the parameterized version using the option ExtraParameters The following options can be given ExtraParameters specify some extra parameters for which the equation has to be solved EXAMPLES angl QSolveRational g 2 f qx f ala 30 x 1 f x q ExtraParameters a0 a1 vs fl Cle CB i on 1 4 al gt Cl nol r2 nso7 QSolveRational q 3 qx 1 f q72 a 2q 2 x 1 f qz x a f az q 6 2q 3 1 z 2 q 5 2q 3 1 f x q omtson f 2 gt C 1 g CHI Chloe C 1 Aa q a 432 too oa qx q2 3 x a 1 a 1 2 q4 2 SEE ALSO DSolvePolynomial DSolveRational QSolvePolynomial RSolvePolynomial RSolveRational 77 RandomPolynomial RandomPolynomial var deg c gives a dense random polynomial in the variable s var of degree deg with integer coefficients between c and c MORE INFORMATION The first argument var can be a single variable for univariate random polyno mials or a list of variables for multivariate random polynomials The second argument deg can be a natu
80. nnihilator LegendreP 2k 1 x x 2 Der z S k hasi Timing rel First FindRelation ann Eliminate z out 132 80 169 Null np33j ByteCount rel ou 33 118360 np134j Support rel onpa De SQ Di Se DE Se Da DP Se D Sp DP Sk D So Da DE Se De Se D Se DE Sk De K DE R K DS DER a Ge EE Dz Se Se Se Sk The next example shows how relations for basis functions can be found that are needed in finite element methods In this instance we are looking for a relation that involves the function itself and its first derivative both can be arbitrarily shifted and whose coefficients are free of x and y hasi Factor FindRelation Annihilator 1 x i LegendreP 2 y 1 x 1 JacobiP j 2i 1 0 2z 1 S z S j Der x Eliminate gt x y Pattern gt OH omprasj 2i j 5 2i 25 5 8 97 D j 3 2i 2j 5 6 D 2 2i 3 i j 38 5 D 2 2i 1 i j 3 87 D 2 j 3 2i 2j 5 2i 2j TISS 4 1 2i 25 7 SD 2 i j 3044 23 5 9i 4 25 78 2i j 3 2i 24 70504 SEE ALSO Annihilator FindCreativeTelescoping FindSupport OreGroebner Basis 40 FindSupport FindSupport ann opts computes only the support of a relation in the ideal ann that satisfies the constraints given by opts MORE INFORMATION The input ann is a list of OrePolynomial expressions that have
81. nomial expressions that constitute a Gr bner basis with respect to the Ore algebra and monomial order that they are represented in and the output is a natural number Note that the Gr bner basis condition is not tested and if violated the result may be wrong What internally happens is that only the leading exponent vectors are considered so alternatively a list of such can be given as input If the ideal ann is O finite then its dimension is 0 EXAMPLES nig Annihilator ChebyshevT n x S n Der z owng nS 1 27 D nz x 1 D2 xD n nprj AnnihilatorDimension oui U npsj Annihilator StirlingS1 k m S k S m Annihilator nondf The expression StirlingS1 k m is not recognized to be O finite The result might not generate a zero dimensional ideal ompa S S k Sm 1 nug AnnihilatorDimension out 19j 1 npo Annihilator StirlingS1 k m StirlingS2 k n S k S m S n Annihilator nondf The expression StirlingS1 k m is not recognized to be O finite The result might not generate a zero dimensional ideal Annihilator nondf The expression StirlingS2 k n is not recognized to be O finite The result might not generate a zero dimensional ideal owpoj Sk Sm Sn kn k Sm Sn kSm n 151 1 au AnnihilatorDimension out 21 2 SEE ALSO Annihilator UnderTheStaircase AnnihilatorSingularities AnnihilatorSingula
82. nput Gr bner basis and m is the number of deltas Modulus 0 All computations are done modulo this number In connection with the option Mode Modular setting Modulus p1 Pn computes the result n times modulo all the prime numbers p OrePolynomialSubstitute If a list of rules a ag b bo is given then the result is computed with these substitutions the variables a b must not be summation or in tegration variables In connection with the option Mode gt Modular one can give a list of such substitutions a gt a1 b gt b1 a gt a2 b gt bo Then the computation is done for each of these substitutions Ansatz Automatic To be used in connection with Mode Modular and the value of this option is best produced using Mode FindSupport Variables To be used in connection with Mode Modular and the value of this option is best produced using Mode FindSupport DenominatorName None To be used in connection with Mode Modular and the value of this option is best produced using Mode FindSupport FileNames gt When using Mode Modular then this option can give a StringForm to specify the locations where all the results have to be stored Each variable in the substitution list see option OrePolynomialSubstitute requires a wild card and an additional wild card is needed if several prime numbers are given in opt
83. nt opoly oupss 1 1 1 1 np159 LeadingExponent 0 opoly ou isg 00 00 00 00 SEE ALSO ChangeMonomialOrder ChangeOreAlgebra LeadingCoefficient LeadingPowerProduct LeadingTerm NormalizeCoefficients OrePolynomial OrePolynomialList Coefficients ToOrePolynomial Support 46 LeadingPower Product LeadingPowerProduct opoly gives the leading power product of the Ore polynomial opoly MORE INFORMATION The input opoly has to be an OrePolynomial expression What is considered as the leading term depends on the Ore algebra in which opoly is presented and of course on the monomial order The leading power product of the zero polynomial is not defined LeadingPowerProduct returns Indeterminate in this case EXAMPLES np160j opoly ToOrePolynomial 1 x72 xx Der a 2 x xx Der 4x 3 OreAlgebra Der z Out 160 1 x D xD Az n161 LeadingPowerProduct opoly Out 161 D m163 opoly ChangeOreAlgebra opoly OreAlgebra z Der x Out 162 r D 423 xD D2 ieai LeadingPowerProduct opoly Out 163 x D nps4 Annihilator StruveH n x S n Der owpeg 2 D 2nz x S 2nzD n n 4 2 xS D n 1 Sn x 2nz 3r 8 4n 10n 2 6 S 2 D 3nz 3x w sst LeadingPowerProduct Q owies D Sn Dr Se SEE ALSO
84. nt is either an annihilating ideal i e a Gr bner basis of such an ideal or any mathematical expression In the latter case Annihilator is internally called with expr The second argument delta indicates whether a summation or an integration problem has to be solved it is then S a 1 or Der a respectively where a is the summation resp integration variable For q summation use QS qa a 1 The third argument ops specifies the surviving Ore operators i e the operators that occur in the principal part P as well as in Q The output consists of two lists the first one containing all the principle parts such that they constitute a Gr bner basis and the second one containing the corresponding delta parts Since the principle of creative telescoping is really one of the main aspects of this package we want to explain shortly what is behind When we want to do a definite sum of the form s f k w then we search for creative telescoping operators that annihilate f and that are of the form T P w Ow S 1 Q k w Sk Ow where Oy stands for some Ore operators that act on the variables w The operator P is called the principal part and Q is called the delta part With such an operator T we can immediately derive a relation for the definite sum T k w Sk Ow e f k w b k a b k a b P w Ow ka f k w Kb 5 9 1 Q k w Sk Ow ka f k w k a b P w dw e V f k w Q k w S Ow e f s w
85. os OreSigma QS z q n oupog 1 x gt g m n c1 amp npo7 ToOrePolynomial g n q 3 QS z q n q 2n q n 1 OreAlgebra QS z q n oupsri x q Seq qx a m295 ApplyOreOperator f x g q n ost La q la gla a a Fla gla wei Annihilator QBinomial n k all owpssi qk q qn Sn a a ak qn ak qqk gk Sk g ak qn Insoo Apply OreOperator QBinomial n k q Out 300 e ger QBinomial n 1 k q C q QBinomial n k q 2 d QBinomial n k 1 q a a QBinomial n k d iani FunctionExpand n 3 L k 2 omtson a ETC d 1 4 1 4 q q crac ail aoai Expand out 302j 0 0 SEE ALSO ApplyOreOperator Delta Der Euler OreAction OreDelta OreSigma S ToOrePolynomial 75 QSolvePolynomial QSolvePolynomial leqn fle q determines whether the linear q shift equation eqn in f r with polynomial coefficients has polynomial solutions and in the affirmative case computes them MORE INFORMATION A q shift equation is an equation that involves f r flax f q x etc It may be given as an equation with head Equal or as the left hand side expression which is then understood to be equal to zero If the coefficients of eqn are rational functions it is multiplied by their common denominator The algorithm works b
86. ped in 5 The output is of the form t c where t is the telescoper and c the certificate the output format is the same as in CreativeTelescoping The following options can be given Return Automatic returns by default the above form t c telescoper returns t and is faster since the certificate is not computed at all and bound returns an integer the precomputed bound for the order of the telescoper EXAMPLES past HermiteTelescoping Exp ry z 2y y 22 2 Der y Der owpas 27D a 2 2y 0 pani HermiteTelescoping E zy 10 y Der y Der z Return bound Out 146 9 SEE ALSO CreativeTelescoping FindCreativeTelescoping 44 LeadingCoefficient LeadingCoefficient opoly gives the leading coefficient of the Ore polynomial opoly MORE INFORMATION The input opoly has to be an OrePolynomial expression What is considered as the leading coefficient depends on the Ore algebra in which opoly is pre sented and of course on the monomial order The leading coefficient of the zero polynomial is not defined LeadingCoefficient returns Indeterminate in this case EXAMPLES np47 opoly ToOrePolynomial 1 x72 xx Der a 2 x xx Der 4x 3 us I T 149 opoly ChangeOreAlgebra opoly OreAlgebra z Der x 149 x D 42 D D 150 LeadingCoefficient opoly OreAlgebra Der z
87. properties algebraic sub stitution for continuous variables and rational linear substitution for dis crete variables p 27 DFiniteTimes ann anna executes the closure property product for the finite functions described by ann and ann p 29 DFiniteTimesHyper ann expr executes the closure property product when one factor expr is hypergeometric and hyperexponential in the re spective variables p 30 DSolvePolynomial leqn f determines whether the ordinary linear differ ential equation eqn in f with polynomial coefficients has polynomial solutions and in the affirmative case computes them p 31 DSolveRational leqn f x determines whether the ordinary linear differential equation eqn in f with polynomial coefficients has rational solutions and in the affirmative case computes them p 32 Euler x represents the Euler operator 6 xD p 33 FGLM gb order transforms the noncommutative Gr bner basis gb into a Grobner basis with respect to the given order p 34 FindCreativeTelescoping expr deltas ops finds creative telescoping rela tions for expr by ansatz p 36 FindRelation ann opts finds relations with certain properties in the annihi lating ideal annby ansatz p 39 FindSupport ann opts computes only the support of a relation in the anni hilating ideal ann that satisfies the given constraints p 41 GBEqual gb gb2 whether two Gr
88. r we mean a single symbol that has been introduced by an Ore extension In particular we do not mean an operator in the sense of a recurrence or a differential equation those are Ore polynomials involving some Ore operators EXAMPLES np2 OreOperators S n 2 S k 2 x Der v S n ouf221 Der x S k S n np22 OreOperators D f zx TE np23 Annihilator Fibonacci n zl onp 2nS x 4 Dy n x a 4 D2 3zD 1 n waat OreOperators oupzj Der x S n SEE ALSO OreDelta OreOperatorQ OreSigma 61 OrePlus OrePlus lopolyi opoly2 computes the sum of the Ore polynomials opoly opoly2 etc OrePlus opoly opoly2 alg translates opoly1 opoly2 etc into the Ore algebra alg and then computes their sum MORE INFORMATION The input polynomials can be either given as OrePolynomial expressions or as standard Mathematica polynomials OrePlus then tries to figure out which Ore algebra is best suited for representing the output Alternatively this algebra can be explicitely given as the last argument To make the work with Ore polynomials more convenient we have defined an upvalue for addition This means that the standard notation opoly opoly2 can be used for addition of Ore polynomials if at least one of the summands is of type OrePolynomial then OrePlus will be called EXAMPLES The o
89. ra generated by the O1 Oa In this sense ann is an annihilating ideal i e a list of OrePolynomial expres sion that form a Gr bner basis in some Ore algebra O and opoly is an operator in O which can be given either as an OrePolynomial expression or as a plain polynomial in the Ore operators of O Note that the dimension of the vector space under the stairs of the resulting Gr bner basis is always smaller or equal than the one of the input ann This fact is particularly useful when a sum of two expressions can be written as an operator applied to a single expression then usually DFiniteOreAction delivers bigger annihilating ideals than DFinitePlus see the example below This command is called by Annihilator if its input contains D or Apply Ore Operator The following options can be given MonomialOrder None specifies the monomial order in which the output should be given None means that the monomial order of the input is taken EXAMPLES Sine and cosine have the same differential equation ieo DF initeOreAction ToOrePolynomial Der x 2 1 Der z Out 60 D 1 The next example shows a situation where DfiniteOreAction is preferable to DFinitePlus find an annihilating ideal for Pn 1 x Pa n51 DFiniteOreAction Annihilator LegendreP n x S n Der 1 S n 1 owe 2n 6n 4 S 2na 2n 32 3 D 2n z 5ng n 3x 1 a 1 D2 x 1
90. ral number and then specifies the total degree of the output or a list of positive integers specifying the degree with respect to each variable The third argument c bounds the size of the integer coefficients The output is a standard Mathematica polynomial and not an OrePolynomial EXAMPLES nsosj RandomPolynomiial 5 1076 obra 9541932 132684x 9497822 5975692 171223x 881334 Insog RandomPolynomial z y z 2 1000 owpoj llz 755ay 877xz 840x 351y 136yz 386y 1092 2112 486 In 310 VE IL 2 Ser Out 310 432 z 3_ Alay 2 142y 2 Say 4892 V Toyz KE 84xz 4 65x2 30vz 42x Ty 2 54y 2 449 z 31y 91yz 95yz 5Tyz 65y 352 592 6z 48 In 311 Federer 2 S a Der z 3 1 oui a Der z a S a a L S a Der 2 S a Der 2 aDer z aDer 2 azDer z z S a zS a azS a S a aS a S a aS a az a Der z Ders zDer z Der z 2 z n 12 ToOrePolynomial OreAlgebra S a Der z owing S 52 D D a z 1 82 Sa D Ca 1 D2 2a az a 2 8 a az a4 z DDR sa z SEE ALSO OrePolynomial ToOrePolynomial 78 RSolvePolynomial RSolvePolynomial leqn f n determines whether the linear recurrence equation egn in f n with poly nomial coefficients has polynomial solutions
91. rities ann start computes the set of singular points in the positive region above start for a system ann of multivariate recurrences MORE INFORMATION The input ann has to be a system of multivariate recurrences given as a list of OrePolynomial expressions They need not to form a Grobner basis but note that it will not be recognized if they happen to be inconsistent since only their leading terms are taken into account The second argument start a list of integers gives the start point of the sequence Then all singular points in the positive region above start i e start N where d is the number of variables are determined By singular points we mean those points where none of the recurrences can be applied either because it would involve values from outside the region or because its leading coefficient vanishes The output is a list of pairs each of which consists of a list of substitutions and a condition under which these substitutions give rise to singular points It is tacitly assumed that the recurrence variables take only integer values hence this condition is not extra stated in the output The region under consideration can be further restricted by giving assumptions on the variables The following options can be given Assumptions gt Assumptions further restrictions on the variables of the recurrences EXAMPLES mp3 AnnihilatorSingularities ToOrePolynomial n 5 S n
92. s OreAlgebraOperators al 9 gives the list of Ore operators that are contained in the Ore algebra alg MORE INFORMATION An Ore operator is an expression for which OreSigma and OreDelta are de fined e g S n or Der z are predefined Ore operators The list returned by OreAlgebraOperators is a subset of that one returned by OreAlgebraGen erators EXAMPLES m95 alg OreAlgebra p Der xz n S n Delta m x oupos K m p n z Ds 1 D Sn O Ami Sm Am npo7 OreAlgebraOperators alg ou 157 Der z S n Delta m SEE ALSO OreAlgebra OreAlgebraGenerators OreAlgebraPolynomial Variables OreOperatorQ OreOperators 54 OreAlgebraPolynomial Variables OreAlgebraPolynomial Variables al 9 gives the list of variables that occur polynomially in the Ore algebra alg MORE INFORMATION Every generator of the Ore algebra alg that is not an Ore operator i e for which OreSigma and OreDelta are not defined is called a polynomial variable EXAMPLES m5 alg OreAlgebra p Der x n S n Delta m x ouf198 K m p n z Ds 1 Dz Sn Sn 0 Am Sm Am ino OreAlgebraPolynomialVariables alg out 199j p n x SEE ALSO OreAlgebra OreAlgebraGenerators OreAlgebraOperators 55 OreDelta OreDelta op defines the endomorphism 6 for the Ore operator op MORE INFORMATION Ore operators like S
93. sed so in total 800 files will be stored in the given directory The reconstruction itself has to be done by other means this functionality is not provided by FindCreativeTelescoping np27 ans FindCreativeTelescoping ann S k 1 Mode FindSupport n p128j FindCreativeTelescoping ann S k 1 Mode gt Modular Sequence Q9 ans FileNames temp fct 1 2 m OrePolynomialSubstitute Table n n0 n0 20 99 Modulus Table NextPrime 2 31 i i 103 SEE ALSO Annihilator CreativeTelescoping FindRelation Printlevel 38 FindRelation FindRelation ann opts computes relations with certain properties that have to be specified by the options opts in the annihilating ideal ann MORE INFORMATION The input ann is a list of OrePolynomial expressions that have to form a Gr bner basis this property is not checked by FindRelation It then makes an ansatz with undetermined coefficients in order to find elements in ann that obey the constraints that have been given in opts In particular this com mand can be used for elimination e g when executing Zeilberger s slow algo rithm 12 and it is usually faster than elimination via Gr bner basis compu tation The following options can be given Eliminate forces the coefficients to be free of the given variables note that this option refers only to the coefficients not to the generators of the Ore a
94. sis elements differ only by sign Nevertheless it is very useful since this test cannot be done by using Equal or SameQ EXAMPLES pagi gbl Annihilator Sum Binomial n k Binomial m n k n k 0 n S m S n oups m 1 Sm n US m n n 2 82 2m UR n 1 np39 gb2 Annihilator Sum 2 k Binomial m k Binomial n k k 0 n S m S n oupa m 1 Sm n US m n n 2 82 2m US n 1 nhat gbl gb2 out 149 False np4 FullSimplify gb1 gb2 ompa 0 2 n 2 C4m 2 8 2 n 1 0 0 mp43 GBEqual gb1 gb2 Out 142 True np143j gb3 FGLM gb2 Lexicographic owpas n 2 8 2m 1 S n 1 m 1 Sm n 1 S m n mp44 GBEqual gb1 gb3 out 144j False P a SEE ALSO FGLM OreGroebnerBasis OrePolynomial 43 HermiteTelescoping HermiteTelescoping expr Der y Derri computes telescoper and certificate for a hyperexponential function expr using the algorithm Hermite telescoping HermiteTelescoping ann Der y Der z computes a creative telescoping relation in the annihilating ideal ann which must be the annihilator of a hyperexponential function MORE INFORMATION For general information about creative telescoping see CreativeTelescoping The algorithm Hermite telescoping has been develo
95. the output Alternatively this algebra can be explicitely given as the last argument To make the work with Ore polynomials more convenient we have defined an upvalue for multiplication This means that the notation opoly opoly2 can be used for multiplying two Ore polynomials if at least one of the factors is of type OrePolynomial then OreTimes will be called The same works for the commutative Times But since Mathematica might reorder the factors this should only be used if the factors in fact commute e g in 2 opoly1 EXAMPLES npsn OreTimes Der z x oupez 7D 1 The following does not work since none of the factors is of type OrePolynomial In 288 Der a xk L oupss Der z x This is another way how to do it ips ToOrePolynomial Der x 0ut 289 Dy inoo sc ae out 290 7D 1 Shift a recurrence by 1 npo ToOrePolynomial 2 4 n h 2 4 n J 3 2n h 1 n 4 14 2 n n h n ompa n 2 52 2n 3 S n 1 ipo S n se owpez n 3 2n 5 S n 2 8 SEE ALSO OrePolynomial OrePolynomialSubstitute OrePlus OrePower ToOrePolynomial 73 Printlevel Printlevel n activates and controls verbose mode displaying information about the cur rent computation up to recursion level n EXAMPLES mpo3 ann Flatten Annihilator LegendreP n x amp S n Der z owpss n 2 82 2nz 3
96. to form a Gr bner basis this property is not checked by FindSupport It then makes an ansatz with undetermined coefficients in order to find elements in ann that obey the constraints that have been given in opts In contrast to FindRelation it computes only the support of such elements In fact FindSupport is called by FindRelation if the support is not specified Especially for bigger examples one might be interested in the support of the result before it is actually computed in order to get a feeling for its size On the other hand FindSupport can be used to make statements of the form there exists no k free recurrence of order up to 100 use Printlevel for switching to verbose mode in order to see which supports have unsuccessfully been tried FindSupport uses homomorphic images for fastly obtaining its results In unlucky cases this can lead to fake solutions The following options can be given Eliminate forces the coefficients to be free of the given variables note that this option refers only to the coefficients not to the generators of the Ore algebra ModuleBasis when ann is a Grobner basis over a module give here a list of natural num bers indicating the location of the position variables among the generators of the Ore algebra Pattern gt include only monomials into the ansatz whose exponent vectors match the given pattern Modulus 2147483629 the prime number with respect to w
97. ts The command DSolvePolynomial is able to deal with parameters these have to occur linearly in the inhomogeneous part Call the parameterized version using the option ExtraParameters The following options can be given ExtraParameters specify some extra parameters for which the equation has to be solved EXAMPLES hog DSolvePolynomial f r 5 f ouftog f x gt C 1 5a np10 DSolvePolynomial z 2f r f x 2x 1 f a 2274 cx 3 3272 3z f zr ExtraParameters c ouo f x gt r 3z 3 c gt 7 SEE ALSO DSolveRational QSolvePolynomial QSolveRational RSolvePolynomial RSolveRational 31 DSolveRational DSolveRational eqn fc determines whether the ordinary linear differential equation egn in f x has rational solutions and in the affirmative case computes them MORE INFORMATION The first argument egn can be given either as an equation with head Equal or as the left hand side expression which is then understood to be equal to zero If the coefficients of eqn are rational functions it is multiplied by their common denominator The second argument is the function to be solved for Following Abramov s algorithm 1 first the denominator of the solution is determined Then DSolvePolynomial is called to find the numerator polynomial The following options can be given ExtraParameters
98. us for demonstrating that a recurrence found be cre ative telescoping need not to be the minimal one that exists for the sum in this instance the sum for k from 0 to n evaluates to 3 and hence has a first order recurrence In a4 CreativeTelescoping 1 k Binomial n k Binomial 3k n Delta k S n Out 44 fim 6 S 15n 21 S 9n 9 2n 3 27K 27k n 27k 9kn 18kn 6k n 3n 2n n 2 k2 2kn 3k n 3n 2 SEE ALSO Annihilator FindCreativeTelescoping HermiteTelescoping SolveCoupledSystem Takayama 16 Delta Delta n represents the forward difference delta operator with respect to n MORE INFORMATION When this operator occurs in an OrePolynomial object it is displayed as An The symbol Delta receives its meaning from the definitions of OreSigma OreDelta and OreAction EXAMPLES masi OreDelta Delta n owas 1 n 5n 41 z1 amp nug ApplyOreOperator Delta k k outasj k 1 k You can use the command ChangeOreAlgebra to switch between the delta and the shift representation of a recurrence maz ToOrePolynomial Delta n 3 OreAlgebra Delta n owun A agl ChangeOreAlgebra OreAlgebra S n ons Sp 38 3S 1 SEE ALSO ApplyOreOperator Der Euler OreAction OreDelta OreSigma QS S ToOrePolynomial 17 Der Der a represents t
99. us usually does n Annihilator D BesselJ n x x Der x oup n a z D2 3n z 3 D n 2n z n a 27 nj expr D BesselJ n x x Out B 5 BesselJ 1 n x BesselJ 1 n z npj Annihilator expr Der z owp z D 623 D 2n a 224 52 D 2n z 62 x D n 2n a 2n z 22 1 The following more advanced example is taken from 8 formula 7 322 r Lea 2 300 2 1 dr FCD 5 F n 2v Inge ab 0 chr n I v eo v In this example the inhomogeneous parts are so complicated that Mathematica needs some little help to get them simplified Hence we use the option Inho mogeneous npoj 1 lhs inhom Annihilator Integrate 2a x v 1 2 GegenbauerC n v x a 1 E bx 2 0 2a Der a Der b S n S v Assumptions gt v gt 1 Inhomogeneous True The annihilating ideal for the left hand side is nice but the inhomogeneous part is so big that we don t want to display it here In 11 lhs oup an 2anv 2an 2av a S 2bvS bn 4bnv bn Ab 2bv D 2b v S abn Aabnv abn Aabv 2abv n 4n v n An 2nv an 4anv an 4a 2av Da 2b v S abn Aabnv abn 4abv 2abv n 6n v n 12nv 4nv 8v 4v A02 Ab v An v 20n v 24n v 32n 76nr 44nv 16 56v
100. utput of OrePlus is always an OrePolynomial expression sometimes as here in output line 2 this cannot be directly be seen np25j OrePlus Der xx x 1 x xx Der x OreAlgebra z Der zx Out 225 De 1 wood Der a Out 226 1 ash Head out 227 OrePolynomial Observe how the representation format of the coefficients here in factored form is preserved by OrePlus np28j ToOrePolynomial n 2 n S n n 2 1 Out 228 n n S n 1 in229 Factor out 229j n n 1 S n 1 n 1 isan OrePlus S n 1 n out 230j N DAS n n 1 SEE ALSO OrePolynomial OrePolynomialSubstitute OrePower OreTimes ToOrePolynomial 62 OrePolynomial OrePolynomial data alg order is the internal representation of an Ore polynomial in the Ore algebra alg with monomial order order MORE INFORMATION The above format is the FullForm representation of an Ore polynomial Thanks to the command ToOrePolynomial you never have to type it explicitely Unless you type FullForm you also will never see this representation since Ore polynomials are displayed in a very nice format each term is displayed as CUY 95 g where c is the coefficient and the g are the generators of the algebra their order is preserved Additionally the terms are ordered according to the monomial order i e the leading term is always in front The first argument
101. y determining a degree bound and then making an ansatz for the solution with undetermined coefficients The command QSolvePolynomial is able to deal with parameters these have to occur linearly in the inhomogeneous part Call the parameterized version using the option ExtraParameters The following options can be given ExtraParameters specify some extra parameters for which the equation has to be solved EXAMPLES anat QSolvePolynomial f qx q710 f x 0 f x q Out 303 f x gt CU anal QSolvePolynomial f q 2 x f gc fle q474 92 1 2 4 3 q 2 gQ 37 f x q Out 304 f x q Lar a insosj QSolvePolynomial f q 32 f r c2 3 f z q ExtraParameters c oapo I f x gt CI e gt C 1 4 1 SEE ALSO DSolvePolynomial DSolveRational QSolveRational RSolvePolynomial RSolveRational 76 QSolveRational QSolveRational leqn fle q determines whether the linear q shift equation eqn in f r with polynomial coefficients has rational solutions and in the affirmative case computes them MORE INFORMATION A q shift equation is an equation that involves f x f qx Toi etc It may be given as an equation with head Equal or as the left hand side expression which is then understood to be equal to zero If the coefficients of eqn are rational functions it is multiplied by their common denomi

HolonomicFunctions (User's Guide)

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