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On the $q$-partial differential equations and $q$-series (1805.02132v1)
Published 6 May 2018 in math.AP and math.AC
Abstract: Using the theory of functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, then, it can be expanded in terms of the product of the Rogers-Szeg\H{o} polynomials. This expansion theorem allows us to develop a general method for proving $q$-identities. A general $q$-transformation formula is derived, which implies Watson's $q$-analog of Whipple's theorem as a special case. A multilinear generating function for the Rogers-Szeg\H{o} polynomials is given. The theory of $q$-exponential operator is revisited.