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