<<RISC`fastZeil`

Fast Zeilberger Package version 3.61
written by Peter Paule, Markus Schorn, and Axel Riese
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

?Gosper

Example 78:

Gosper[k k!,k]

Gosper[k k!,{k,0,n}]

The factorial does not have a hypergeometric antidifference:

Gosper[k!,k]

The harmonic numbers are not hypergeometric:

Gosper[1/k,{k,1,n}]

Homework example:

Gosper[Binomial[2k,k]α^k,{k,0,n}]

Gosper[Binomial[2k,k](1/4)^k,{k,0,n}]

Gosper[Binomial[x+k,k],{k,0,n}]

Gosper[Binomial[n,k],{k,0,n}]

?Zb

Zb[Binomial[n,k],{k,0,n},n]

Zb[Binomial[n,k],{k,0,Infinity},n]

Zb[Binomial[n,k],k,n]

Prove[]

Zb[Binomial[n,k]^2,{k,0,Infinity},n]

Zb[(-1)^k Binomial[n,k]/Binomial[x+k,k],k,n]

Jacobi polynomials

Zb[Pochhammer[-n,k] Pochhammer[n+α+β+1,k]/Pochhammer[α+1,k]/k! ((1-x)/2)^k,{k,0,n},n]

Non-minimality of Zeilberger’s algorithm:

d=1;
Zb[(-1)^k Binomial[n,k] Binomial[d k,n],{k,0,n},n]

d=2;
Zb[(-1)^k Binomial[n,k] Binomial[d k,n],{k,0,n},n]

d=3;
rec=Zb[(-1)^k Binomial[n,k] Binomial[d k,n],{k,0,n},n]

FullForm[rec]

rec = ReleaseHold[rec[[1]]]

a[n_]:=(-d)^n

rec1 = rec/.SUM->a

FullSimplify[rec1]

<<RISC`Guess`

HolonomicFunctions Package version 1.7.3 (21-Mar-2017)
written by Christoph Koutschan
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

Package GeneratingFunctions version 0.8 written by Christian Mallinger
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

Guess Package version 0.52
written by Manuel Kauers
Copyright Research Institute for Symbolic Computation (RISC),
Johannes Kepler University, Linz, Austria

data = Table[Sum[(-1)^k Binomial[n,k] Binomial[d k,n],{k,0,n}],{n,0,50}];

GuessMinRE[data,s[n]]