A multiple q-exponential differential operational identity

Abstract

Using Hartogs’ fundamental theorem for analytic functions in several complex variables and q-partial differential equations, we establish a multiple q-exponential differential formula for analytic functions in several variables. With this identity, we give new proofs of a variety of important classical formulas including Bailey’s ₆ψ₆ series summation formula and the Atakishiyev integral. A new transformation formula for a double q-series with several interesting special cases is given. A new transformation formula for a ₃ψ₃ series is proved.