Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions
In this paper, we perform a further investigation on the Apostol–Bernoulli and Apostol–Euler functions introduced by Luo. By using the Fourier expansions of the Apostol–Bernoulli and Apostol–Euler polynomials, we establish some symmetric identities for the Apostol–Bernoulli and Apostol–Euler functio...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/7/1384 |
| _version_ | 1797587382171598848 |
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| author | Yuan He |
| author_facet | Yuan He |
| author_sort | Yuan He |
| collection | DOAJ |
| description | In this paper, we perform a further investigation on the Apostol–Bernoulli and Apostol–Euler functions introduced by Luo. By using the Fourier expansions of the Apostol–Bernoulli and Apostol–Euler polynomials, we establish some symmetric identities for the Apostol–Bernoulli and Apostol–Euler functions. As applications, some known results, for example, Raabe’s multiplication formula and Hermite’s identity, are deduced as special cases. |
| first_indexed | 2024-03-11T00:36:15Z |
| format | Article |
| id | doaj.art-73cb5dc1c0844496969e299d2c40a9ab |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T00:36:15Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-73cb5dc1c0844496969e299d2c40a9ab2023-11-18T21:34:20ZengMDPI AGSymmetry2073-89942023-07-01157138410.3390/sym15071384Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler FunctionsYuan He0School of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, ChinaIn this paper, we perform a further investigation on the Apostol–Bernoulli and Apostol–Euler functions introduced by Luo. By using the Fourier expansions of the Apostol–Bernoulli and Apostol–Euler polynomials, we establish some symmetric identities for the Apostol–Bernoulli and Apostol–Euler functions. As applications, some known results, for example, Raabe’s multiplication formula and Hermite’s identity, are deduced as special cases.https://www.mdpi.com/2073-8994/15/7/1384Apostol–Bernoulli functionsBernoulli functionsApostol–Euler functionsquasi-periodic Euler functionscombinatorial identity |
| spellingShingle | Yuan He Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions Symmetry Apostol–Bernoulli functions Bernoulli functions Apostol–Euler functions quasi-periodic Euler functions combinatorial identity |
| title | Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions |
| title_full | Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions |
| title_fullStr | Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions |
| title_full_unstemmed | Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions |
| title_short | Applications of Symmetric Identities for Apostol–Bernoulli and Apostol–Euler Functions |
| title_sort | applications of symmetric identities for apostol bernoulli and apostol euler functions |
| topic | Apostol–Bernoulli functions Bernoulli functions Apostol–Euler functions quasi-periodic Euler functions combinatorial identity |
| url | https://www.mdpi.com/2073-8994/15/7/1384 |
| work_keys_str_mv | AT yuanhe applicationsofsymmetricidentitiesforapostolbernoulliandapostoleulerfunctions |