A telescoping method for double summations

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 = Δ i ( R 1 F )...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2006-11, Vol.196 (2), p.553-566
Main Authors: Chen, William Y.C., Hou, Qing-Hu, Mu, Yan-Ping
Format: Article
Language:English
Subjects:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Computer science
control theory
systems
Double summation
Exact sciences and technology
Hypergeometric term
Mathematical analysis
Mathematics
Sciences and techniques of general use
Special functions
Theoretical computing
Zeilberger's algorithm
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Summary: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 = Δ i ( R 1 F ) + Δ 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, Carlitz's identities, the Apéry–Schmidt–Strehl identity, the Graham–Knuth–Patashnik identity, and the Petkovšek–Wilf–Zeilberger identity.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2005.10.010