Some expansion formulas for q-series and their applications

In this paper, we establish some general expansion formulas for q-series. Three of Liu's identities motivate us to search and find such type of formulas. These expansion formulas include as special cases or limiting cases many q-identities including the q-Gauss summation formula, the q-Pfaff-Sa...

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Bibliographic Details
Published in:Journal of combinatorial theory. Series A 2025-01, Vol.209, p.105941, Article 105941
Main Authors: He, Bing, Wen, Suzhen
Format: Article
Language:English
Subjects:
Basic hypergeometric series
Dilcher's identity
Dirichlet L-function
Double Rogers-Ramanujan type identity
Expansion formula
Hurwitz zeta function
Orthogonality relation
Summation formula
Transformation formula
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Summary:In this paper, we establish some general expansion formulas for q-series. Three of Liu's identities motivate us to search and find such type of formulas. These expansion formulas include as special cases or limiting cases many q-identities including the q-Gauss summation formula, the q-Pfaff-Saalschütz summation formula, three of Jackson's transformation formulas and Sears' terminating ϕ34 transformation formula. As applications, we provide a new proof of the orthogonality relation for continuous dual q-Hahn polynomials, establish some generating functions for special values of the Dirichlet L-functions and the Hurwitz zeta functions, give extensions of three of Liu's identities, establish an extension of Dilcher's identity, and deduce various double Rogers-Ramanujan type identities.
ISSN:0097-3165
DOI:10.1016/j.jcta.2024.105941