Burcu SilindirAhmet Yantir2025-10-062021035451800354-51802406-093310.2298/FIL2111855Shttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85126292238&doi=10.2298%2FFIL2111855S&partnerID=40&md5=0a16f48be8fb4cca9aa4e104de2e74fchttps://gcris.yasar.edu.tr/handle/123456789/9034In 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<inf>(qh)</inf> 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. © 2022 Elsevier B.V. All rights reserved.English(q H)-analytic Functions, (q H)-integral, (qh)-burger’s Equation, (qh)-gauss’s Binomial Formula, (qh)diffusion Equation, (qh)trigonometric Functions, Additive Property Of (qh-exponential FunctionsArticle