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

Abstract

Using the method of the q-exponential differential operator, we give an extension of the Sears ₄𝜙₃ 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.