A telescoping method for double summations

摘要

We present a method to prove hypergeometric double summation identities. Given a hypergeometric term F(n,i,j), we aim to find a difference operator L=a0(n)N0+a1(n)N1+⋯+ar(n)Nr and rational functions R1(n,i,j),R2(n,i,j) such that LF=Δi(R1F)+Δj(R2F). Based on simple divisibility considerations, we show that the denominators of R1 and R2 must possess certain factors which can be computed from F(n,i,j). Using these factors as estimates, we may find the numerators of R1 and R2 by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews–Paule identity, Carlitz's identities, the Apéry–Schmidt–Strehl identity, the Graham–Knuth–Patashnik identity, and the Petkovšek–Wilf–Zeilberger identity.