Apéry-Type Series and Colored Multiple Zeta Values
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866917851495923712 |
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| author | Xu, Ce Zhao, Jianqiang |
| author_facet | Xu, Ce Zhao, Jianqiang |
| contents | In this paper, we introduce and study new classes of Apéry-type series involving the multiple $t$-harmonic sums by combining the methods of iterated integral and Fourier--Legendre series expansions, where the multiple $t$-harmonic sums are a variation of multiple harmonic sums in which all the summation indices are restricted to odd numbers only. Our approach also enables us to generalize some old classes of Apéry-type series involving harmonic sums to those with products of harmonic sums and multiple $t$-harmonic sums. In all of these series, the central binomial coefficients appear as $a_n^{\pm 1}$ or $a_n^{\pm 2}$ where $a_n=\binom{2n}{n}/4^n$. We show that every such series can be expressed as either the real or the imaginary part of a $\mathbb Q$-linear combination of colored multiple zeta values of level 4. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_10998 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Apéry-Type Series and Colored Multiple Zeta Values Xu, Ce Zhao, Jianqiang Number Theory In this paper, we introduce and study new classes of Apéry-type series involving the multiple $t$-harmonic sums by combining the methods of iterated integral and Fourier--Legendre series expansions, where the multiple $t$-harmonic sums are a variation of multiple harmonic sums in which all the summation indices are restricted to odd numbers only. Our approach also enables us to generalize some old classes of Apéry-type series involving harmonic sums to those with products of harmonic sums and multiple $t$-harmonic sums. In all of these series, the central binomial coefficients appear as $a_n^{\pm 1}$ or $a_n^{\pm 2}$ where $a_n=\binom{2n}{n}/4^n$. We show that every such series can be expressed as either the real or the imaginary part of a $\mathbb Q$-linear combination of colored multiple zeta values of level 4. |
| title | Apéry-Type Series and Colored Multiple Zeta Values |
| topic | Number Theory |
| url | https://arxiv.org/abs/2111.10998 |