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 functionsBased 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.
The Boros-Moll polynomials arise in the evaluation of a quartic integral. The original double summation formula does not imply the fact that the coefficients of these polynomials are positive. Boros and Moll proved the positivity by using Ramanujan's Master Theorem to reduce the double sum to a single sum. Based on the structure of reluctant functions introduced by Mullin and Rota along with an extension of Foata's bijection between Meixner endofunctions and bi-colored permutations, we find a combinatorial proof of the positivity. In fact, from our combinatorial argument one sees that it is essentially the binomial theorem that makes it possible to reduce the double sum to a single sum.
Given two polynomials, we find a convergence property of the GCD of the rising factorial and the falling factorial. Based on this property, we present a unified approach to computing the universal denominators as given by Gosper's algorithm and Abramov's algorithm for finding rational solutions to linear difference equations with polynomial coefficients. Our approach easily extends to the q-analogues.
Abstract. We present an operator approach to Rogers-type formulas and Mehler's formulas for the Al-Salam-Carlitz polynomials U n (x, y, a; q). By using the q-exponential operator, we obtain a Rogers-type formula which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler's formula due to by Al-Salam and Carlitz, which requires a terminating condition on a 3 φ 2 series. By means of the Cauchy companion augmentation operator, we obtain Mehler's formula in a similar form, but it does not need the terminating condition. We also give several identities on the generating functions for products of the Al-Salam-Carlitz polynomials which are extensions of formulas for Rogers-Szegö polynomials.Keywords: Al-Salam-Carlitz polynomial, the q-exponential operator, the homogeneous qshift operator, the Cauchy companion operator, the Rogers-type formula, Mehler's formula
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