Abstract
We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series. We demonstrate that a Lagrange interpolation formula always leads to very-well-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced \({_{4}\phi_{3}}\) to a very-well-poised \({_{8}\phi_{7}}\) is equivalent to the Rodrigues-type formula for the Askey-Wilson polynomials. By applying the Leibniz formula for the Askey-Wilson operator we also establish the \({_{8}\phi_{7}}\) summation theorem.
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This work was done while Ismail was at the Department of Mathematics, City University of Kong Kong. His research was supported by NPST Program of King Saud University, project number 10-MAT1293-02, and by Grants Council of Hong Kong under contract # 101411.
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Ismail, M.E.H., Stanton, D. Some Combinatorial and Analytical Identities. Ann. Comb. 16, 755–771 (2012). https://doi.org/10.1007/s00026-012-0158-1
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DOI: https://doi.org/10.1007/s00026-012-0158-1