Recurrence Relations for $β(2k)$ and $ζ(2k + 1)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915470655881216 |
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| author | Kyrion, Tobias |
| author_facet | Kyrion, Tobias |
| contents | In this work we study integrals of the form $\int_{0}^{\infty}\frac{\tanh(x)}{x}\mathrm{sech}(x)^{L}\exp(-Tx)dx$. For $L \in \mathbb{N}$, $L \leq 4$ and $T \in \mathbb{R}$ we give explicit expressions in terms of derivatives of the Hurwitz zeta function at negative integers. We use these expressions to evaluate these integrals for $T \in \mathbb{N}_{0}$ exactly. For the special case $T = 0$ we give explicit evaluations for any $L \in \mathbb{N}$ based on the functional equations for $β(s)$ and $ζ(s)$. As it turns out the value of $\int_{0}^{\infty}\frac{\tanh(x)}{x}\mathrm{sech}(x)^{2N + 1}dx$ is a linear combination of $β(2)$, ..., $β(2N + 2)$ and the value of $\int_{0}^{\infty}\frac{\tanh(x)}{x}\mathrm{sech}(x)^{2N}dx$ a linear combination of $ζ(3)$, ..., $ζ(2N + 1)$. We give recursive formulae for the coefficients in these linear combinations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11643 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recurrence Relations for $β(2k)$ and $ζ(2k + 1)$ Kyrion, Tobias General Mathematics In this work we study integrals of the form $\int_{0}^{\infty}\frac{\tanh(x)}{x}\mathrm{sech}(x)^{L}\exp(-Tx)dx$. For $L \in \mathbb{N}$, $L \leq 4$ and $T \in \mathbb{R}$ we give explicit expressions in terms of derivatives of the Hurwitz zeta function at negative integers. We use these expressions to evaluate these integrals for $T \in \mathbb{N}_{0}$ exactly. For the special case $T = 0$ we give explicit evaluations for any $L \in \mathbb{N}$ based on the functional equations for $β(s)$ and $ζ(s)$. As it turns out the value of $\int_{0}^{\infty}\frac{\tanh(x)}{x}\mathrm{sech}(x)^{2N + 1}dx$ is a linear combination of $β(2)$, ..., $β(2N + 2)$ and the value of $\int_{0}^{\infty}\frac{\tanh(x)}{x}\mathrm{sech}(x)^{2N}dx$ a linear combination of $ζ(3)$, ..., $ζ(2N + 1)$. We give recursive formulae for the coefficients in these linear combinations. |
| title | Recurrence Relations for $β(2k)$ and $ζ(2k + 1)$ |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2508.11643 |