  Using the method of $q$-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 application of this transformation formula
to Rogers--Hecke type series are discussed. More than 100 Rogers--Hecke type
identities including Andrews's identities for the sums of three squares and the
sums of three triangular numbers are obtained.
