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

Abstract

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