Overview - Maple Help

Overview of the SumTools:-Hypergeometric Subpackage

 Calling Sequence SumTools:-Hypergeometric:-command(arguments) command(arguments)

Description

 • The SumTools:-Hypergeometric subpackage provides tools for finding closed forms of definite and indefinite sums of hypergeometric type. It can also be used for certifying and proving combinatorial identities. The subpackage consists of three main components:
 - Normal forms of rational functions and of hypergeometric terms: MultiplicativeDecomposition, PolynomialNormalForm, RationalCanonicalForm, SumDecomposition
 - Algorithms for definite and indefinite sums of hypergeometric type: ExtendedGosper, ExtendedZeilberger, Gosper, IsZApplicable, KoepfGosper, KoepfZeilberger, LowerBound, MinimalZpair, Zeilberger, ZeilbergerRecurrence, ZpairDirect
 - Applications: DefiniteSum, IndefiniteSum, WZMethod
 • Other commands that deal with hypergeometric terms include: AreSimilar, ConjugateRTerm, EfficientRepresentation, IsHolonomic, IsHypergeometricTerm, IsProperHypergeometricTerm, RegularGammaForm, Verify
 • Each command in the SumTools:-Hypergeometric subpackage can be accessed by using either the long form or the short form of the command name in the command calling sequence.
 • The long form, SumTools:-Hypergometric:-command, is always available. The short form can be used after loading the package.

List of SumTools:-Hypergeometric Subpackage Commands

 The following is a list of available commands.

 To display the help page for a particular Hypergeometric command, see Getting Help with a Command in a Package.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}:-\mathrm{Hypergeometric}\right)$
 $\left[{\mathrm{AreSimilar}}{,}{\mathrm{BottomSequence}}{,}{\mathrm{CanonicalRepresentation}}{,}{\mathrm{ConjugateRTerm}}{,}{\mathrm{DefiniteSum}}{,}{\mathrm{DefiniteSumAsymptotic}}{,}{\mathrm{EfficientRepresentation}}{,}{\mathrm{ExtendedGosper}}{,}{\mathrm{ExtendedZeilberger}}{,}{\mathrm{Gosper}}{,}{\mathrm{IndefiniteSum}}{,}{\mathrm{IsHolonomic}}{,}{\mathrm{IsHypergeometricTerm}}{,}{\mathrm{IsProperHypergeometricTerm}}{,}{\mathrm{IsZApplicable}}{,}{\mathrm{KoepfGosper}}{,}{\mathrm{KoepfZeilberger}}{,}{\mathrm{LowerBound}}{,}{\mathrm{MinimalTelescoper}}{,}{\mathrm{MinimalZpair}}{,}{\mathrm{MultiplicativeDecomposition}}{,}{\mathrm{PolynomialNormalForm}}{,}{\mathrm{RationalCanonicalForm}}{,}{\mathrm{RegularGammaForm}}{,}{\mathrm{SumDecomposition}}{,}{\mathrm{Verify}}{,}{\mathrm{WZMethod}}{,}{\mathrm{Zeilberger}}{,}{\mathrm{ZeilbergerRecurrence}}{,}{\mathrm{ZpairDirect}}\right]$ (1)

Definite sum example:

 > $T≔{\mathrm{binomial}\left(2n,2k\right)}^{2}$
 ${T}{≔}{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{2}{}{k}}\right)}^{{2}}$ (2)
 > ${\sum }_{k=0}^{n}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}T=\mathrm{DefiniteSum}\left(T,n,k,0..n\right)$
 ${\sum }_{{k}{=}{0}}^{{n}}{}{\left(\genfrac{}{}{0}{}{{2}{}{n}}{{2}{}{k}}\right)}^{{2}}{=}\frac{{\left({-1}\right)}^{{n}}{}\left(\genfrac{}{}{0}{}{{2}{}{n}}{{n}}\right)}{{2}}{+}\frac{\left(\genfrac{}{}{0}{}{{4}{}{n}}{{2}{}{n}}\right)}{{2}}$ (3)

Construct the Apery's recurrence.

 > $T≔{\mathrm{binomial}\left(n,k\right)}^{2}{\mathrm{binomial}\left(n+k,k\right)}^{2}$
 ${T}{≔}{\left(\genfrac{}{}{0}{}{{n}}{{k}}\right)}^{{2}}{}{\left(\genfrac{}{}{0}{}{{n}{+}{k}}{{k}}\right)}^{{2}}$ (4)
 > $\mathrm{lre}≔\mathrm{ZeilbergerRecurrence}\left(T,n,k,a,0..n\right)$
 ${\mathrm{lre}}{≔}\left({{n}}^{{3}}{+}{3}{}{{n}}^{{2}}{+}{3}{}{n}{+}{1}\right){}{a}{}\left({n}\right){+}\left({{n}}^{{3}}{+}{6}{}{{n}}^{{2}}{+}{12}{}{n}{+}{8}\right){}{a}{}\left({n}{+}{2}\right){+}\left({-}{34}{}{{n}}^{{3}}{-}{153}{}{{n}}^{{2}}{-}{231}{}{n}{-}{117}\right){}{a}{}\left({n}{+}{1}\right){=}{0}$ (5)

Replace n by $n-1$ in $\mathrm{lre}$.

 > $\mathrm{collect}\left(\mathrm{subs}\left(n=n-1,\mathrm{lre}\right),\left[a\left(n+1\right),a\left(n\right),a\left(n-1\right)\right],'\mathrm{factor}'\right)$
 ${\left({n}{+}{1}\right)}^{{3}}{}{a}{}\left({n}{+}{1}\right){-}\left({2}{}{n}{+}{1}\right){}\left({17}{}{{n}}^{{2}}{+}{17}{}{n}{+}{5}\right){}{a}{}\left({n}\right){+}{{n}}^{{3}}{}{a}{}\left({n}{-}{1}\right){=}{0}$ (6)

The above recurrence equation is required in the proof of the irrationality of Zeta(3).

References

 Abramov, S.A.; Geddes, K.O.; and Le, H.Q. "Computer Algebra Library for the Construction of the Minimal Telescopers." Proceedings of ICMS'2002, pp. 319-329. World Scientific, 2002.
 Le, H.Q.; Abramov, S.A.; and Geddes, K.O. "HypergeometricSum: A Maple Package for Finding Closed Forms of Indefinite and Definite Sums of Hypergeometric Type." Technical Report CS-2001-24. Ontario: Department of Computer Science, University of Waterloo, 2001.