A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function
In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functio...
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| Format: | Article |
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MDPI AG
2021-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/3/233 |
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| author | Daeyeoul Kim Yilmaz Simsek |
| author_facet | Daeyeoul Kim Yilmaz Simsek |
| author_sort | Daeyeoul Kim |
| collection | DOAJ |
| description | In this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials. |
| first_indexed | 2024-03-09T03:43:45Z |
| format | Article |
| id | doaj.art-b77d0fef6e5b42fb963b19114a3fe10a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T03:43:45Z |
| publishDate | 2021-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b77d0fef6e5b42fb963b19114a3fe10a2023-12-03T14:37:19ZengMDPI AGMathematics2227-73902021-01-019323310.3390/math9030233A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta FunctionDaeyeoul Kim0Yilmaz Simsek1Department of Mathematics and Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju 54896, KoreaDepartment of Mathematics, Faculty of Science, University of Akdeniz, Antalya TR-07058, TurkeyIn this paper, we further study the generating function involving a variety of special numbers and ploynomials constructed by the second author. Applying the Mellin transformation to this generating function, we define a new class of zeta type functions, which is related to the interpolation functions of the Apostol–Bernoulli polynomials, the Bernoulli polynomials, and the Euler polynomials. This new class of zeta type functions is related to the Hurwitz zeta function, the alternating Hurwitz zeta function, and the Lerch zeta function. Furthermore, by using these functions, we derive some identities and combinatorial sums involving the Bernoulli numbers and polynomials and the Euler numbers and polynomials.https://www.mdpi.com/2227-7390/9/3/233Bernoulli numbers and polynomialsEuler numbers and polynomialsApostol–Bernoulli and Apostol–Euler numbers and polynomialsHurwitz–Lerch zeta functionHurwitz zeta functionalternating Hurwitz zeta function |
| spellingShingle | Daeyeoul Kim Yilmaz Simsek A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function Mathematics Bernoulli numbers and polynomials Euler numbers and polynomials Apostol–Bernoulli and Apostol–Euler numbers and polynomials Hurwitz–Lerch zeta function Hurwitz zeta function alternating Hurwitz zeta function |
| title | A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function |
| title_full | A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function |
| title_fullStr | A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function |
| title_full_unstemmed | A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function |
| title_short | A New Family of Zeta Type Functions Involving the Hurwitz Zeta Function and the Alternating Hurwitz Zeta Function |
| title_sort | new family of zeta type functions involving the hurwitz zeta function and the alternating hurwitz zeta function |
| topic | Bernoulli numbers and polynomials Euler numbers and polynomials Apostol–Bernoulli and Apostol–Euler numbers and polynomials Hurwitz–Lerch zeta function Hurwitz zeta function alternating Hurwitz zeta function |
| url | https://www.mdpi.com/2227-7390/9/3/233 |
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