Some Combinatorial and Analytical Identities

Mourad E.H. Ismail; Dennis Stanton

doi:10.1007/s00026-012-0158-1

Some Combinatorial and Analytical Identities

Mourad E.H. Ismail, Dennis Stanton

School of Mathematics

Research output: Contribution to journal › Article › peer-review

17 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 language	English (US)
Pages (from-to)	755-771
Number of pages	17
Journal	Annals of Combinatorics
Volume	16
Issue number	4
DOIs	https://doi.org/10.1007/s00026-012-0158-1
State	Published - 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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title = "Some Combinatorial and Analytical Identities",

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.",

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author = "Ismail, {Mourad E.H.} and Dennis Stanton",

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.",

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AU - Stanton, Dennis

N1 - 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.

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N2 - 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.

AB - 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.

KW - Dilcher

KW - Fu and Lascoux

KW - Lagrange type interpolation

KW - Prodinger and Uchimura

KW - Summation theorems

KW - Watson transformation

KW - bibasic sums

KW - identities of Chen and Liu

KW - partitions

KW - polynomial expansions

KW - the Gasper identity

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JO - Annals of Combinatorics

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ER -

Some Combinatorial and Analytical Identities

Abstract

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