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On a Transformation of Triple $q$-Series and Rogers-Hecke Type Series (2401.14099v2)
Published 25 Jan 2024 in math.CV, math.CO, math.NT, and math.QA
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.