Generalized quantum exponential function and its applications

Abstract

This article aims to present (q, h)-analogue of exponential function which unifies, extends hand 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.

Gauss’s binomial formula and additive property of exponential functions on T(q,h)

“…In this section, we first briefly summarize the calculus on (q, h)-time scales that we presented in [18]. In [6], a two-parameter time scale T (q,h) is defined by…”

“…Proposition 2.2. [18] For the arbitrary functions f and , the product and the quotient rules are given by…”

“…Exponential function (28) recovers many exponential functions studied in the literature. (i) For γ = 1 and x 0 = 0, (28) reduces to the (q, h)-exponential function exp (q,h) (αx), introduced in [18] Exp (q,h) (α…”

“…For that reason, a special discrete time scale T (q,h) which unifies and extends hand q-analysis, is presented in [6] in order to study (q, h)-fractional calculus. Since then a variety of mathematicians have contributed to the (q, h)-analysis such as (q, h)-analogue of Laplace transform [16], (q, h)-analogue of quantum splines [8] and (q, h)-analogue of binomials and the wave equation [18].…”

“…The main objective of the current article is to overcome some of the deficiencies in front of the applicability of time scales. In Section 2, we improve (q, h)-calculus that we presented in [18]. The emphasize is placed on (q, h)-analogue of integration.…”