Some Combinatorial and Analytical Identities

Mourad E.H. Ismail, Dennis Stanton

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)755-771
Number of pages17
JournalAnnals of Combinatorics
Volume16
Issue number4
DOIs
StatePublished - Dec 2012

Bibliographical note

Funding Information:
∗ 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.

Keywords

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

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