|
Abstract: 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)N 0 + a1(n)N 1 + ··· +
ar(n)N r and
rational functions R1(n, i, j), R2(n, i, j) such that LF =  i(R1F) +
j(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, the Carlitz's identities, the Apéry-
Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and
the Petkovšek-Wilf-Zeilberger identity.
Keywords:
Zeilberger's algorithm, double summation, hypergeometric term.
AMS Classification: 33F10, 68W30.
Download: pdf
|