<< zb.m

Fast Zeilberger Package by Peter Paule and Markus Schorn (enhanced by Axel Riese) — © RISC Linz — V 3.54 (02/23/05)

?Gosper

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

Prove[]

(* Harmonic numbers are not hypergeometric: *)

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

Gosper[Pochhammer[a,k]/k!,{k,0,n}]

Gosper[(-1)^k Binomial[m,k],{k,0,n}]

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

Gosper[1/(k^2+Sqrt[5] k-1),{k,0,n}]

Gosper[(4k-1)/(2k-1)^2 16^(-k) Binomial[2k,k]^2,{k,0,n}]

(* Zeilberger's algorithm *)

?Zb

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

(* note that a HoldForm[...] is wrapped around SUM which needs to be released if we want to solve the recurrence above *)

FullForm[rec]

rec = ReleaseHold[rec/.SUM->g]

RSolve[rec,g[n],n]

(* Using Zeilberger's algorithm to derive the three term recurrence for Jacobi polynomials: *)

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

Prove[]

(* Three term recurrence for Laguerre polynomials: *)

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