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

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.