Title

Some Combinatorial And Analytical Identities

Keywords

bibasic sums; Dilcher; Fu and Lascoux; identities of Chen and Liu; Lagrange type interpolation; partitions; polynomial expansions; Prodinger and Uchimura; Summation theorems; the Gasper identity; Watson transformation

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 4φ 3 to a very-well-poised 8φ 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 8φ 7 summation theorem. © 2012 Springer Basel.

Publication Date

12-1-2012

Publication Title

Annals of Combinatorics

Volume

16

Issue

4

Number of Pages

755-771

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1007/s00026-012-0158-1

Socpus ID

84870381643 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84870381643

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