Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus
Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the co...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2022-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2022/7969503 |
| _version_ | 1826820118811770880 |
|---|---|
| author | Nabiullah Khan Mohd Ghayasuddin Dojin Kim Junesang Choi |
| author_facet | Nabiullah Khan Mohd Ghayasuddin Dojin Kim Junesang Choi |
| author_sort | Nabiullah Khan |
| collection | DOAJ |
| description | Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised. |
| first_indexed | 2024-03-13T03:34:27Z |
| format | Article |
| id | doaj.art-74acb394dcd245fb8c09d8cb2ca3b1f4 |
| institution | Directory Open Access Journal |
| issn | 2314-4785 |
| language | English |
| last_indexed | 2025-02-16T06:24:30Z |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj.art-74acb394dcd245fb8c09d8cb2ca3b1f42025-02-03T06:47:31ZengWileyJournal of Mathematics2314-47852022-01-01202210.1155/2022/7969503Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral CalculusNabiullah Khan0Mohd Ghayasuddin1Dojin Kim2Junesang Choi3Department of Applied MathematicsDepartment of MathematicsDepartment of MathematicsDepartment of MathematicsAmong a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised.http://dx.doi.org/10.1155/2022/7969503 |
| spellingShingle | Nabiullah Khan Mohd Ghayasuddin Dojin Kim Junesang Choi Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus Journal of Mathematics |
| title | Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus |
| title_full | Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus |
| title_fullStr | Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus |
| title_full_unstemmed | Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus |
| title_short | Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus |
| title_sort | two new generalizations of extended bernoulli polynomials and numbers and umbral calculus |
| url | http://dx.doi.org/10.1155/2022/7969503 |
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