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

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Názov: Some special functions and cylindrical diffusion equation on α-time scale
Autori: Silindir Burcu, Tuncer Zehra, Gergün Seçil, Yantir Ahmet
Zdroj: Demonstratio Mathematica, Vol 58, Iss 1, Pp 51-68 (2025)
Informácie o vydavateľovi: De Gruyter, 2025.
Rok vydania: 2025
Zbierka: LCC:Mathematics
Predmety: exponential function, gamma function, bessel equations, bessel functions, cylindrical diffusion equation, 26e70, 34n05, 33d05, 33c10, 33b15, 34b30, Mathematics, QA1-939
Popis: This article is dedicated to present various concepts on α\alpha -time scale, including power series, Taylor series, binomial series, exponential function, gamma function, and Bessel functions of the first kind. We introduce the α\alpha -exponential function as a series, examine its absolute and uniform convergence, and establish its additive identity by employing the α\alpha -Gauss binomial formula. Furthermore, we define the α\alpha -gamma function and prove α\alpha -analogue of the Bohr-Mollerup theorem. Specifically, we demonstrate that the α\alpha -gamma function is the unique logarithmically convex solution of f(s+1)=ϕ(s)f(s)f\left(s+1)=\phi \left(s)f\left(s), f(1)=1f\left(1)=1, where ϕ(s)\phi \left(s) refers to the α\alpha -number. In addition, we present Euler’s infinite product form and asymptotic behavior of α\alpha -gamma function. As an application, we propose α\alpha -analogue of the cylindrical diffusion equation, from which α\alpha -Bessel and modified α\alpha -Bessel equations are derived. We explore the solutions of the α\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 α\alpha -analogues of exponential and gamma functions and investigate their reductions to discrete and ordinary counterparts.
Druh dokumentu: article
Popis súboru: electronic resource
Jazyk: English
ISSN: 2391-4661
Relation: https://doaj.org/toc/2391-4661
DOI: 10.1515/dema-2025-0131
Prístupová URL adresa: https://doaj.org/article/b99a7ab5f2684ccaad5b0f76e5e3bb6f
Prístupové číslo: edsdoj.b99a7ab5f2684ccaad5b0f76e5e3bb6f
Databáza: Directory of Open Access Journals
Popis
Abstrakt:This article is dedicated to present various concepts on α\alpha -time scale, including power series, Taylor series, binomial series, exponential function, gamma function, and Bessel functions of the first kind. We introduce the α\alpha -exponential function as a series, examine its absolute and uniform convergence, and establish its additive identity by employing the α\alpha -Gauss binomial formula. Furthermore, we define the α\alpha -gamma function and prove α\alpha -analogue of the Bohr-Mollerup theorem. Specifically, we demonstrate that the α\alpha -gamma function is the unique logarithmically convex solution of f(s+1)=ϕ(s)f(s)f\left(s+1)=\phi \left(s)f\left(s), f(1)=1f\left(1)=1, where ϕ(s)\phi \left(s) refers to the α\alpha -number. In addition, we present Euler’s infinite product form and asymptotic behavior of α\alpha -gamma function. As an application, we propose α\alpha -analogue of the cylindrical diffusion equation, from which α\alpha -Bessel and modified α\alpha -Bessel equations are derived. We explore the solutions of the α\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 α\alpha -analogues of exponential and gamma functions and investigate their reductions to discrete and ordinary counterparts.
ISSN:23914661
DOI:10.1515/dema-2025-0131