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arXiv · 2401.14099

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

Zhi-Guo Liu
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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.

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Zhi-Guo Liu. 2024-01-25. On a Transformation of Triple $q$-Series and Rogers-Hecke Type Series. https://doi.org/10.3842/sigma.2024.086

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