Annals of Combinatorics

, Volume 16, Issue 4, pp 755–771 | Cite as

Some Combinatorial and Analytical Identities

  • Mourad E. H. Ismail
  • Dennis Stanton


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.

Mathematics Subject Classification

05A19 33D15 05A30 33D70 


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 


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Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Central FloridaOrlando, FloridaUSA
  2. 2.Department of MathematicsCollege of Science, King Saud UniversityRiyadhSaudi Arabia
  3. 3.School of Mathematics, College of Science and EngineeringUniversity of MinnesotaMinneapolisUSA

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