New Results Involving Riemann Zeta Function Using Its Distributional Representation
The relation of special functions with fractional integral transforms has a great influence on modern science and research. For example, an old special function, namely, the Mittag–Leffler function, became the queen of fractional calculus because its image under the Laplace transform is known to a l...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-05-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/5/254 |
| _version_ | 1797499754701127680 |
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| author | Asifa Tassaddiq Rekha Srivastava |
| author_facet | Asifa Tassaddiq Rekha Srivastava |
| author_sort | Asifa Tassaddiq |
| collection | DOAJ |
| description | The relation of special functions with fractional integral transforms has a great influence on modern science and research. For example, an old special function, namely, the Mittag–Leffler function, became the queen of fractional calculus because its image under the Laplace transform is known to a large audience only in this century. By taking motivation from these facts, we use distributional representation of the Riemann zeta function to compute its Laplace transform, which has played a fundamental role in applying the operators of generalized fractional calculus to this well-studied function. Hence, similar new images under various other popular fractional transforms can be obtained as special cases. A new fractional kinetic equation involving the Riemann zeta function is formulated and solved. Thereafter, a new relation involving the Laplace transform of the Riemann zeta function and the Fox–Wright function is explored, which proved to significantly simplify the results. Various new distributional properties are also derived. |
| first_indexed | 2024-03-10T03:51:58Z |
| format | Article |
| id | doaj.art-2907a8947ecd4b5993b22e8ddf399fc4 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T03:51:58Z |
| publishDate | 2022-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-2907a8947ecd4b5993b22e8ddf399fc42023-11-23T11:03:35ZengMDPI AGFractal and Fractional2504-31102022-05-016525410.3390/fractalfract6050254New Results Involving Riemann Zeta Function Using Its Distributional RepresentationAsifa Tassaddiq0Rekha Srivastava1Department of Basic Sciences and Humanities, College of Computer and Information Sciences, Majmaah University, Al Majmaah 11952, Saudi ArabiaDepartment of Mathematics and Statistics, University of Victoria, Victoria, BC V8P 5C2, CanadaThe relation of special functions with fractional integral transforms has a great influence on modern science and research. For example, an old special function, namely, the Mittag–Leffler function, became the queen of fractional calculus because its image under the Laplace transform is known to a large audience only in this century. By taking motivation from these facts, we use distributional representation of the Riemann zeta function to compute its Laplace transform, which has played a fundamental role in applying the operators of generalized fractional calculus to this well-studied function. Hence, similar new images under various other popular fractional transforms can be obtained as special cases. A new fractional kinetic equation involving the Riemann zeta function is formulated and solved. Thereafter, a new relation involving the Laplace transform of the Riemann zeta function and the Fox–Wright function is explored, which proved to significantly simplify the results. Various new distributional properties are also derived.https://www.mdpi.com/2504-3110/6/5/254delta functionRiemann zeta-functionfractional transformsFox–Wright-functiongeneralized fractional kinetic equation |
| spellingShingle | Asifa Tassaddiq Rekha Srivastava New Results Involving Riemann Zeta Function Using Its Distributional Representation Fractal and Fractional delta function Riemann zeta-function fractional transforms Fox–Wright-function generalized fractional kinetic equation |
| title | New Results Involving Riemann Zeta Function Using Its Distributional Representation |
| title_full | New Results Involving Riemann Zeta Function Using Its Distributional Representation |
| title_fullStr | New Results Involving Riemann Zeta Function Using Its Distributional Representation |
| title_full_unstemmed | New Results Involving Riemann Zeta Function Using Its Distributional Representation |
| title_short | New Results Involving Riemann Zeta Function Using Its Distributional Representation |
| title_sort | new results involving riemann zeta function using its distributional representation |
| topic | delta function Riemann zeta-function fractional transforms Fox–Wright-function generalized fractional kinetic equation |
| url | https://www.mdpi.com/2504-3110/6/5/254 |
| work_keys_str_mv | AT asifatassaddiq newresultsinvolvingriemannzetafunctionusingitsdistributionalrepresentation AT rekhasrivastava newresultsinvolvingriemannzetafunctionusingitsdistributionalrepresentation |