Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 086, 37 pages      arXiv:2401.14099      https://doi.org/10.3842/SIGMA.2024.086
Contribution to the Special Issue on Basic Hypergeometric Series Associated with Root Systems and Applications in honor of Stephen C. Milne

On a Transformation of Triple $q$-Series and Rogers-Hecke Type Series

Zhi-Guo Liu
School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education) & Shanghai KeyLaboratory of PMMP, East China Normal University, Shanghai 200241, P.R. China

Received January 26, 2024, in final form September 15, 2024; Published online October 04, 2024

Abstract
Using the method of the $q$-exponential differential operator, we give an extension of the Sears $_4\phi_3$ transformation formula. Based on this extended formula and a $q$-series expansion formula for an analytic function around the origin, we present a transformation formula for triple $q$-series, which includes several interesting special cases, especially a double $q$-series summation formula. Some applications of this transformation formula to Rogers-Hecke type series are discussed. More than 100 Rogers-Hecke type identities including Andrews' identities for the sums of three squares and the sums of three triangular numbers are obtained.

Key words: $q$-partial differential equation; double $q$-series summation; triple $q$-hypergeometric series; $q$-exponential differential operator; Rogers-Hecke type series.

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