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SumTools[Hypergeometric][IsHolonomic] - test if a given bivariate hypergeometric term is holonomic
SumTools[Hypergeometric][IsProperHypergeometricTerm] - test if a given bivariate hypergeometric term is proper
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Calling Sequence
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IsHolonomic(T, n, k)
IsProperHypergeometricTerm(T, n, k)
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Parameters
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T
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hypergeometric term of n and k
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n
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variable
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k
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variable
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Description
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The IsProperHypergeometricTerm(T,n,k) command returns true if  is a proper hypergeometric term. Otherwise, it returns false.
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The IsHolonomic(T,n,k) command returns true if the bivariate hypergeometric term  is holonomic. Otherwise, it returns false.
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It can be shown that  is proper if and only if it is holonomic.
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Note: If  is a proper hypergeometric term, the termination of Zeilberger's algorithm is guaranteed.
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Examples
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![with(SumTools[Hypergeometric])](/support/helpjp/helpview.aspx?si=5491/file06623/math128.png)
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](/support/helpjp/helpview.aspx?si=5491/file06623/math139.png)
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![[-2/Pi^(1/2), factorial(n-k)*factorial(k+4*n)*(-1/4)^k*4^n*factorial(n-k+1/2)*factorial(n+1)/(factorial(n+1-k)*factorial(2+k+4*n)*factorial(k)*factorial(n-2*k+1)*factorial(n))]](/support/helpjp/helpview.aspx?si=5491/file06623/math142.png)
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](/support/helpjp/helpview.aspx?si=5491/file06623/math160.png)
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![[-(54/5)/(Pi^(1/2)*(-5*3^(1/3)+5*3^(1/3)*n-3*k)^3), factorial(24+47*n-5*k)*factorial(n-k)*(-1/4)^k*4^n*factorial(n-k+1/2)*factorial(n+1)/(factorial(23+47*n-5*k)*factorial(n+1-k)*factorial(k)*factorial(n-2*k+1)*factorial(n))]](/support/helpjp/helpview.aspx?si=5491/file06623/math163.png)
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References
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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.
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