Some special functions and cylindrical diffusion equation on α-time scale
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 uni...
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| Format: | Article |
| Language: | English |
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De Gruyter
2025-06-01
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| Series: | Demonstratio Mathematica |
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| Online Access: | https://doi.org/10.1515/dema-2025-0131 |
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| _version_ | 1849693945496338432 |
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| author | Silindir Burcu Tuncer Zehra Gergün Seçil Yantir Ahmet |
| author_facet | Silindir Burcu Tuncer Zehra Gergün Seçil Yantir Ahmet |
| author_sort | Silindir Burcu |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-b99a7ab5f2684ccaad5b0f76e5e3bb6f |
| institution | DOAJ |
| issn | 2391-4661 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Demonstratio Mathematica |
| spelling | doaj-art-b99a7ab5f2684ccaad5b0f76e5e3bb6f2025-08-20T03:20:14ZengDe GruyterDemonstratio Mathematica2391-46612025-06-01581516810.1515/dema-2025-0131Some special functions and cylindrical diffusion equation on α-time scaleSilindir Burcu0Tuncer Zehra1Gergün Seçil2Yantir Ahmet3Department of Mathematics, Dokuz Eylül University, İzmir, TurkeyThe Graduate School of Natural and Applied Sciences, Dokuz Eylül University, İzmir, TurkeyDepartment of Mathematics, Dokuz Eylül University, İzmir, TurkeyDepartment of Mathematics, Yaşar University, İzmir, TurkeyThis 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.https://doi.org/10.1515/dema-2025-0131exponential functiongamma functionbessel equationsbessel functionscylindrical diffusion equation26e7034n0533d0533c1033b1534b30 |
| spellingShingle | Silindir Burcu Tuncer Zehra Gergün Seçil Yantir Ahmet Some special functions and cylindrical diffusion equation on α-time scale Demonstratio Mathematica exponential function gamma function bessel equations bessel functions cylindrical diffusion equation 26e70 34n05 33d05 33c10 33b15 34b30 |
| title | Some special functions and cylindrical diffusion equation on α-time scale |
| title_full | Some special functions and cylindrical diffusion equation on α-time scale |
| title_fullStr | Some special functions and cylindrical diffusion equation on α-time scale |
| title_full_unstemmed | Some special functions and cylindrical diffusion equation on α-time scale |
| title_short | Some special functions and cylindrical diffusion equation on α-time scale |
| title_sort | some special functions and cylindrical diffusion equation on α time scale |
| topic | exponential function gamma function bessel equations bessel functions cylindrical diffusion equation 26e70 34n05 33d05 33c10 33b15 34b30 |
| url | https://doi.org/10.1515/dema-2025-0131 |
| work_keys_str_mv | AT silindirburcu somespecialfunctionsandcylindricaldiffusionequationonatimescale AT tuncerzehra somespecialfunctionsandcylindricaldiffusionequationonatimescale AT gergunsecil somespecialfunctionsandcylindricaldiffusionequationonatimescale AT yantirahmet somespecialfunctionsandcylindricaldiffusionequationonatimescale |