Some special functions and cylindrical diffusion equation on α -time scale

This article is dedicated to present various concepts on α -time scale, including power series, Taylor series, binomial series, exponential function, gamma function, and Bessel functions of the first kind. We introduce the α -exponential function as a series, examine its absolute and uniform convergence, and establish its additive identity by employing the α -Gauss binomial formula. Furthermore, we define the α -gamma function and prove α -analogue of the Bohr-Mollerup theorem. Specifically, we demonstrate that the α -gamma function is the unique logarithmically convex solution of f ( s + 1 ) = ϕ ( s ) f ( s ) , f ( 1 ) = 1 , where ϕ ( s ) refers to the α -number. In addition, we present Euler’s infinite product form and asymptotic behavior of α -gamma function. As an application, we propose α -analogue of the cylindrical diffusion equation, from which α -Bessel and modified α -Bessel equations are derived. We explore the solutions of the α -cylindrical diffusion equation using the separation of variables technique, revealing analogues of the Bessel and modified Bessel functions of order zero of the first kind. Finally, we illustrate the graphs of the α -analogues of exponential and gamma functions and investigate their reductions to discrete and ordinary counterparts.