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

Abstract

In this article, we focus our attention on (q,h)-Gauss?s binomial formula from which we discover the additive property of (q; h)-exponential functions. We state the (q,h)-analogue of Gauss?s binomial formula in terms of proper polynomials on T(q,h) which own essential properties similar to ordinary polynomials. We present (q,h)-Taylor series and analyze the conditions for its convergence. We introduce a new (q,h)-analytic exponential function which admits the additive property. As consequences, we study (q,h)-hyperbolic functions, (q,h)-trigonometric functions and their significant properties such as (q,h)-Pythagorean Theorem and double-angle formulas. Finally, we illustrate our results by a first order (q,h)-difference equation, (q,h)-analogues of dynamic diffusion equation and Burger?s equation. Introducing (q,h)-Hopf-Cole transformation, we obtain (q,h)-shock soliton solutions of Burger?s equation.