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SumTools[Hypergeometric][ExtendedZeilberger] - construct a minimal Z-pair
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Calling Sequence
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ExtendedZeilberger(V, n, k, En)
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Parameters
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V
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definite sum of hypergeometric term
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n
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name
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k
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name
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En
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name denoting the shift operator with respect to n
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Description
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It can be shown that a Z-pair for  exists if and only if a Z-pair for the hypergeometric term  exists.
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Examples
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![with(SumTools[Hypergeometric])](/support/helpjp/helpview.aspx?si=5476/file06620/math102.png)
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![[2*n+3+(2*n+4)*En^2+(-7-4*n)*En, 2*k^2*(-1)^k*binomial(n+1, k)*binomial(2*k, k)/((-n-2+k)*2^(2*k))]](/support/helpjp/helpview.aspx?si=5476/file06620/math123.png)
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Try the Maple command sum:
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![(1/2)*(4*n+6+2*(-1)^n)*GAMMA(n+3/2)/((n+1)*(n+2)*Pi^(1/2)*GAMMA(n))+(1/2)*(4*n+6+2*(-1)^n)*GAMMA(n+3/2)/(Pi^(1/2)*GAMMA(n+3))+(n+1)*(-(-1)^n+1)*GAMMA(n+3/2)/(Pi^(1/2)*GAMMA(n+3))+sum(k*(-1)^k*(PIECEWISE([1, k = 0], [sin(Pi*k)/(Pi*k), otherwise]))*binomial(2*k, k)/((k+1)*2^(2*k)), k = 0 .. n)](/support/helpjp/helpview.aspx?si=5476/file06620/math148.png)
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References
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Le, H.Q. "Computing the Minimal Telescoper for Sums of Hypergeometric Terms." SIGSAM Bulletin: Communications on Computer Algebra. Vol. 35 No. 3. (2001): 2-10.
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