Some Combinatorial and Analytical Identities

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 verywell-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced 4φ3 to a very-well-poised 8φ7 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 8φ7 summation theorem. Filename: FuLascoux-105. Reference: Annals of Combinatorics, 2011. AMS Subject Classification 2010 Primary: 05A19 and 33D15, Secondary: 05A30 and 33D70.