The computation of $ζ(2k)$, $β(2k+1)$ and beyond by using telescoping series
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929655145037824 |
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| author | Ciaurri, Óscar Navas, Luis M. Ruiz, Francisco J. Varona, Juan L. |
| author_facet | Ciaurri, Óscar Navas, Luis M. Ruiz, Francisco J. Varona, Juan L. |
| contents | We present some simple proofs of the well-known expressions for \[
ζ(2k) = \sum_{m=1}^\infty \frac{1}{m^{2k}},
\qquad
β(2k+1) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^{2k+1}}, \] where $k = 1,2,3,\dots$, in terms of the Bernoulli and Euler polynomials. The computation is done using only the defining properties of these polynomials and employing telescoping series. The same method also yields integral formulas for $ζ(2k+1)$ and $β(2k)$.
In addition, the method also applies to series of type \[
\sum_{m\in\mathbb{Z}} \frac{1}{(2m-μ)^s},
\qquad
\sum_{m\in\mathbb{Z}} \frac{(-1)^m}{(2m+1-μ)^s}, \] in this case using Apostol-Bernoulli and Apostol-Euler polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08063 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The computation of $ζ(2k)$, $β(2k+1)$ and beyond by using telescoping series Ciaurri, Óscar Navas, Luis M. Ruiz, Francisco J. Varona, Juan L. Number Theory Primary 40C15, Secondary 11M06 We present some simple proofs of the well-known expressions for \[ ζ(2k) = \sum_{m=1}^\infty \frac{1}{m^{2k}}, \qquad β(2k+1) = \sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^{2k+1}}, \] where $k = 1,2,3,\dots$, in terms of the Bernoulli and Euler polynomials. The computation is done using only the defining properties of these polynomials and employing telescoping series. The same method also yields integral formulas for $ζ(2k+1)$ and $β(2k)$. In addition, the method also applies to series of type \[ \sum_{m\in\mathbb{Z}} \frac{1}{(2m-μ)^s}, \qquad \sum_{m\in\mathbb{Z}} \frac{(-1)^m}{(2m+1-μ)^s}, \] in this case using Apostol-Bernoulli and Apostol-Euler polynomials. |
| title | The computation of $ζ(2k)$, $β(2k+1)$ and beyond by using telescoping series |
| topic | Number Theory Primary 40C15, Secondary 11M06 |
| url | https://arxiv.org/abs/2307.08063 |