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A Telescoping Algorithm for Double Summations | | William Y.C. Chen
; Qing-Hu Hou
; Yan-Ping Mu
; | | Date: |
26 Apr 2005 | | Subject: | Combinatorics MSC-class: 33F10, 68W30 | math.CO | | Abstract: | We present an algorithm 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 $ L F = Delta_i (R_1 F) + Delta_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 algorithm is valid for the Andrews-Paule identity, the Carlitz’s identities, the Apéry-Schmidt-Strehl identity, the Graham-Knuth-Patashnik identity, and the Petkovv{s}ek-Wilf-Zeilberger identity. | | Source: | arXiv, math.CO/0504525 | | Services: | Forum | Review | PDF | Favorites | |
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