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.