Overview of the SumTools[Hypergeometric] Subpackage - Maple Programming Help

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Overview of the SumTools[Hypergeometric] Subpackage

 

Calling Sequence

Description

List of SumTools[Hypergeometric] Subpackage Commands

Examples

References

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.

• 

Since the underlying implementation of the SumTools[Hypergeometric] subpackage is a module, it is also possible to use the form SumTools:-Hypergeometric:-command or SumTools[Hypergeometric]:-command to access a command. For more information, see Module Members.

List of SumTools[Hypergeometric] Subpackage Commands

  

The following is a list of available commands.

AreSimilar

BottomSequence

ConjugateRTerm

DefiniteSum

DefiniteSumAsymptotic

EfficientRepresentation

ExtendedGosper

ExtendedZeilberger

Gosper

IndefiniteSum

IsHolonomic

IsHypergeometricTerm

IsProperHypergeometricTerm

IsZApplicable

KoepfGosper

KoepfZeilberger

LowerBound

MinimalTelescoper

MinimalZpair

MultiplicativeDecomposition

PolynomialNormalForm

RationalCanonicalForm

RegularGammaForm

SumDecomposition

Verify

WZMethod

Zeilberger

ZeilbergerRecurrence

ZpairDirect

 

 

 

  

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

Examples

withSumTools[Hypergeometric]

AreSimilar,BottomSequence,CanonicalRepresentation,ConjugateRTerm,DefiniteSum,DefiniteSumAsymptotic,EfficientRepresentation,ExtendedGosper,ExtendedZeilberger,Gosper,IndefiniteSum,IsHolonomic,IsHypergeometricTerm,IsProperHypergeometricTerm,IsZApplicable,KoepfGosper,KoepfZeilberger,LowerBound,MinimalTelescoper,MinimalZpair,MultiplicativeDecomposition,PolynomialNormalForm,RationalCanonicalForm,RegularGammaForm,SumDecomposition,Verify,WZMethod,Zeilberger,ZeilbergerRecurrence,ZpairDirect

(1)

Definite sum example:

Tbinomial2n,2k2

Tbinomial2n,2k2

(2)

k=0nT=DefiniteSumT,n,k,0..n

k=0nbinomial2n,2k2=1422n+522Γ2n+12n12Γ2n+124n+121n22n+2Γn+122nπ121n4nΓn+122πΓn+12Γn+14n1

(3)

Construct the Apery's recurrence.

Tbinomialn,k2binomialn+k,k2

Tbinomialn,k2binomialn+k,k2

(4)

lreZeilbergerRecurrenceT,n,k,a,0..n

lren3+3n2+3n+1an+n3+6n2+12n+8an+2+34n3153n2231n117an+1=0

(5)

Replace n by n1 in lre.

collectsubsn=n1,lre,an+1,an,an1,'factor'

n+13an+12n+117n2+17n+5an+n3an1=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.

See Also

help

LREtools

rsolve

sum

SumTools

UsingPackages

with