Burcu SilindirAhmet YantirSilindir, BurcuYantir, Ahmet2025-10-0620210354-51802406-093310.2298/FIL2111855S2-s2.0-85126292238http://dx.doi.org/10.2298/FIL2111855Shttps://gcris.yasar.edu.tr/handle/123456789/6801https://doi.org/10.2298/FIL2111855SIn 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.Englishinfo:eu-repo/semantics/openAccess(q h)-Gauss's binomial formula, (q h)-integral, (q h)-analytic functions, additive property of (q h)-exponential functions, (q h)-trigonometric functions, (q h)-diffusion equation, (q h)-Burger's equation(Q, h)-Diffusion EquationAdditive Property of (q, h)-Exponential Functions(Q, h)-Trigonometric Functions(Q, h)-Burger’s Equation(Q, h)-Analytic Functions(Q,h)-Burger’s Equation(Q,h)Trigonometric FunctionsAdditive Property of (q,h-Exponential Functions(Q,h)-Gauss’s Binomial Formula(Q, h)-Integral(Q,h)Diffusion EquationArticle