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SumTools[Hypergeometric]

  

IsHolonomic

  

test if a given bivariate hypergeometric term is holonomic

  

IsProperHypergeometricTerm

  

test if a given bivariate hypergeometric term is proper

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

IsHolonomic(T, n, k)

IsProperHypergeometricTerm(T, n, k)

Parameters

T

-

hypergeometric term of n and k

n

-

variable

k

-

variable

Description

• 

The IsProperHypergeometricTerm(T,n,k) command returns true if Tn,k is a proper hypergeometric term. Otherwise, it returns false.

• 

The IsHolonomic(T,n,k) command returns true if the bivariate hypergeometric term Tn,k is holonomic. Otherwise, it returns false.

• 

A bivariate hypergeometric term Tn,k is proper if it can be written as Tn,k=Pn,kTpn,k where Pn,k is a polynomial of n and k, and Tpn,k=unvki=1lkbi+nai+gi!i=1mapi+bpi+gpi!, ai,bi,api,bpi are integers, and l,m are non-negative integers, gi,gpi,u,v are complex numbers.

• 

It can be shown that Tn,k is proper if and only if it is holonomic.

  

Note: If Tn,k is a proper hypergeometric term, the termination of Zeilberger's algorithm is guaranteed.

Examples

withSumTools[Hypergeometric]:

T4k4n1k4n+3k4n2k4n1kbinomialn+1,kbinomial2n2k+1,n

T:=4k4n1k4n+3k4n2k4n1kbinomialn+1,kbinomial2n2k+1,n

(1)

ConjugateRTerm[1]T,n,k,'listform'

2π,nk!k+4n!14k4nnk+12!n+1!n+1k!2+k+4n!k!n2k+1!n!

(2)

IsProperHypergeometricTermT,n,k

true

(3)

T4894n+10k1kbinomialn+1,kbinomial2n2k+1,n55n+k9133

T:=4894n+10k1kbinomialn+1,kbinomial2n2k+1,n55n+k91/33

(4)

ConjugateRTerm[1]T,n,k,'listform'

545π531/3n531/33k3,24+47n5k!nk!14k4nnk+12!n+1!23+47n5k!n+1k!k!n2k+1!n!

(5)

IsProperHypergeometricTermT,n,k

false

(6)

References

  

Abramov, S.A., and Petkovsek, M. "Proof of a Conjecture of Wilf and Zeilberger." Preprint series. Vol. 39. (2001): 748. University of Ljubljana, ISSN 1318--4865.

See Also

SumTools[Hypergeometric]

SumTools[Hypergeometric][ConjugateRTerm]

SumTools[Hypergeometric][IsHypergeometricTerm]

SumTools[Hypergeometric][Zeilberger]

 


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