4,604 research outputs found
The Method of Combinatorial Telescoping
We present a method for proving q-series identities by combinatorial
telescoping, in the sense that one can transform a bijection or a
classification of combinatorial objects into a telescoping relation. We shall
illustrate this method by giving a combinatorial proof of Watson's identity
which implies the Rogers-Ramanujan identities.Comment: 11 pages, 5 figures; to appear in J. Combin. Theory Ser.
A Telescoping method for Double Summations
We present a method to prove hypergeometric double summation identities.
Given a hypergeometric term , we aim to find a difference operator and rational functions
such that .
Based on simple divisibility considerations, we show that the denominators of
and must possess certain factors which can be computed from . Using these factors as estimates, we may find the numerators of
and 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 Ap\'ery-Schmidt-Strehl identity, the Graham-Knuth-Patashnik
identity, and the Petkov\v{s}ek-Wilf-Zeilberger identity.Comment: 22 pages. to appear in J. Computational and Applied Mathematic
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