Gauss's binomial formula and additive property of exponential functions on $\mathbb{T}_{(q,h)}$

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 $\mathbb{T}_{(q,h)}$ 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.