FILOMAT, vol.35, no.11, pp.3855-3877, 2021 (SCI-Expanded)
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,T-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.