SumTools[Hypergeometric]
WZMethod
perform Wilf-Zeilberger's algorithm
Calling Sequence
Parameters
Description
Examples
References
WZMethod(f,r,n,k,cert)
f
-
function of n and k
r
function of n
n
variable
k
cert
(optional) name; assigned the computed WZ certificate
The WZMethod(f,r,n,k,cert) command certifies identities of the form ∑kfn,k=rn.
Let Fn,k=fn,krn if rn≠0 and Fn,k=fn,k, otherwise. If the method succeeds in certifying the given identity, the output is a list of two elements F,G representing the WZ-pair F,G such that Fn+1,k−Fn,k=Gn,k+1−Gn,k. Otherwise, it returns the error message "WZ method fails".
If the method is successful and if the fifth optional argument cert is given, cert is assigned the WZ certificate Rn,k=Gn,kFn,k.
It is assumed that for each integer 0≤n, limk→∞Gn,k=0 and limk→−∞Gn,k=0.
withSumToolsHypergeometric:
Proof of Gauss's 2F1 identity:
f≔n+k!b+k!c−n−1!c−b−1!c+k!n−1!c−n−b−1!k+1!b−1!
r≔1
WZpair≔WZMethodf,r,n,k,cert:
F≔WZpair1
F≔n+k!b+k!c−n−1!c−b−1!c+k!n−1!c−n−b−1!k+1!b−1!
G≔WZpair2
G≔−c−n−2−b!n!n+k!c−n−1!−c−n−2!n+1+k!n−1!c−n−b−1!b+k!c−b−1!c+kk+1c−n−b−1!n−1!b−1!k+1!c−n−2−b!n!c+k!bk+bn−ck+nk+b−c+k+n+1
−k+1c+kc−n−1n
Proof of Dixon's identity:
F≔−1kbinomialn+b,n+kbinomialn+c,c+kbinomialb+c,b+k
F≔−1kn+bn+kn+cc+kb+cb+k
r≔n+b+c!n!b!c!
WZpair≔WZMethodF,r,n,k,certificate:
F≔−1kn+bn+kn+cc+kb+cb+kn!b!c!n+b+c!
G≔−n+1+bn+1+kn+1+cc+kn+1!n+b+c!−n+1+b+c!n+bn+kn+cc+kn!c!b!b+cb+k−1kc+kn+1+kb+k2n+b+c!n+1+b+c!bcn+bk2+ck2+k2n+bc+k2
certificate
c+kb+k2−n+k−1n+1+b+c
Wilf, H., and Zeilberger, D. "Rational function certify combinatorial identities." J. Amer. Math. Soc. Vol. 3. (1990): 147-158.
See Also
limit
Sum
SumTools[Hypergeometric][Gosper]
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