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SumTools[Hypergeometric]

  

KoepfZeilberger

  

perform Koepf-Zeilberger's algorithm

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

KoepfZeilberger(T, n, k, En)

Parameters

T

-

(m, l)-fold hypergeometric term in n and k

n

-

name

k

-

name

En

-

name; denote the shift operator with respect to n

Description

• 

For a specified (m, l)-fold hypergeometric term Tn,k in n and k, the KoepfZeilberger(T, n, k, En) command constructs for Tn,k a Z-pair L,G that consists of a linear difference operator with coefficients that are polynomials of n over the complex number field

L=avnEnv+...+a1nEn+a0n

  

and a function Gn,k such that

LTn,k=Gn,k+1Gn,k.

• 

A function Tn,k is an (m, l)-fold hypergeometric term if Tn+m,kTn,k and Tn,k+lTn,k are rational functions of n and k.

• 

The output from the KoepfZeilberger command is a list of two elements L,G representing the computed Z-pair L,G.

Examples

withSumToolsHypergeometric:

Tbinomial2n3,2k

T2n32k

(1)

ZpairKoepfZeilbergerT,n,k,En

ZpairEn34,6kn212k1k2n32kn3+k12n3+2k1n

(2)

VerifyT,Zpair,n,k,En

true

(3)

IsHypergeometricTermT,n

false

(4)

Note that since T is not a hypergeometric term in n, Zeilberger's algorithm is not applicable to T.

References

  

Koepf, W. "Algorithms for m-fold Hypergeometric Summation." Journal of Symbolic Computation. Vol. 20 No. 4. (1995): 399-417.

  

Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.

See Also

SumTools[Hypergeometric]

sumtools[hypersum]

SumTools[KoepfGosper]

SumTools[Zeilberger]