Some special functions and cylindrical diffusion equation on <i>α</i> -time scale

Abstract This article is dedicated to present various concepts on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -time scale, including power series, Taylor series, binomial series, exponential function, gamma function, and Bessel functions of the first kind. We introduce the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -exponential function as a series, examine its absolute and uniform convergence, and establish its additive identity by employing the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -Gauss binomial formula. Furthermore, we define the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -gamma function and prove <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -analogue of the Bohr-Mollerup theorem. Specifically, we demonstrate that the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -gamma function is the unique logarithmically convex solution of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>ϕ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> f\left(s+1)=\phi \left(s)f\left(s) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> f\left(1)=1 , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ϕ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \phi \left(s) refers to the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -number. In addition, we present Euler’s infinite product form and asymptotic behavior of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -gamma function. As an application, we propose <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -analogue of the cylindrical diffusion equation, from which <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> </m:math> \alpha -Bessel and modified <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">