Some Combinatorial and Analytical Identities

    Authors

    M. E. H. Ismail;D. Stanton

    Comments

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    Abbreviated Journal Title

    Ann. Comb.

    Keywords

    partitions; identities of Chen and Liu; Dilcher; Fu and Lascoux; Prodinger and Uchimura; Summation theorems; polynomial expansions; bibasic sums; Watson transformation; the Gasper identity; Lagrange type; interpolation; Q-TAYLOR THEOREMS; DIVISOR FUNCTIONS; INTERPOLATION; FOUNDATIONS; Mathematics, Applied

    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.

    Journal Title

    Annals of Combinatorics

    Volume

    16

    Issue/Number

    4

    Publication Date

    1-1-2012

    Document Type

    Article

    Language

    English

    First Page

    755

    Last Page

    771

    WOS Identifier

    WOS:000311510800008

    ISSN

    0218-0006

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