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-(qT-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.