SumTools[Hypergeometric]
ExtendedZeilberger
construct a minimal Z-pair
Calling Sequence
Parameters
Description
Examples
References
ExtendedZeilberger(V, n, k, En)
V
-
definite sum of hypergeometric term
n
name
k
En
name denoting the shift operator with respect to n
Let Tm,k be a hypergeometric term of m and k. Let Vn,k=∑m=−n+bn+dTm,k where b and d are integers. The ExtendedZeilberger(V, n, k, En) command, an extension to Zeilberger's algorithm, constructs the minimal Z-pair for Vn,k provided that it exists.
It can be shown that a Z-pair for Vn,k exists if and only if a Z-pair for the hypergeometric term Tn,k exists.
withSumToolsHypergeometric:
T≔−1kbinomialm,kbinomial2k,k⋅122k
T≔−1kmk2kk22k
V≔SumT,m=0..n
V≔∑m=0n−1kmk2kk22k
ExtendedZeilbergerV,n,k,En
2n+4En2+−4n−7En+2n+3,2k2−1kn+1k2kk−n+k−222k
_EnvDoubleSum≔true
SumV,k=0..n=DefiniteSumV,n,k,0..n
∑k=0n∑m=0n−1kmk2kk22k=2n+12nn4n
Try the Maple command sum:
sumsumT,m=0..n,k=0..n
4−nn+12n+1n+∑k=0n2sinπk−1k4−k2k−1k−1πk+1
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.
See Also
sum
SumTools[Hypergeometric][DefiniteSum]
SumTools[Hypergeometric][IsZApplicable]
SumTools[Hypergeometric][Zeilberger]
SumTools[Hypergeometric][ZeilbergerRecurrence]
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