A Telescoping Method for Double Summations
William Y.C. Chen, Qing-Hu Hou and Yan-Ping Mu
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 = a_{0}(n)N ^{0} + a_{1}(n)N ^{1} + ··· +
a_{r}(n)N^{ r} and
rational functions R_{1}(n, i, j), R_{2}(n, i, j) such that LF = _{i}(R_{1}F) +
_{j}(R_{2}F). Based on simple divisibility considerations, we show that
the denominators of R_{1} and R_{2} must possess certain factors which can be computed from F(n, i, j). Using these factors as estimates, we may
find the numerators of R_{1} and R_{2} 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. |