Generalized quantum exponential function and its applications

Abstract

This article aims to present $(q,h)$-analogue of exponential function which unifies, extends $h$- and $q$-exponential functions in a convenient and efficient form. For this purpose, we introduce generalized quantum binomial which serves as an analogue of an ordinary polynomial. We state $(q,h)$-analogue of Taylor series  and introduce generalized quantum exponential function which is determined by Taylor series in generalized quantum binomial. Furthermore, we prove existence and uniqueness theorem  for a first order, linear, homogeneous IVP whose solution produces an infinite product form for generalized quantum exponential function. We conclude that both representations of generalized quantum exponential function are equivalent. We illustrate our results by ordinary and partial difference equations. Finally, we present a generic dynamic wave equation which admits generalized trigonometric, hyperbolic type of solutions and produces various kinds of partial differential/difference equations.