Apéry-Like Sums and Colored Multiple Zeta Values
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929608638595072 |
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| author | Xu, Ce Zhao, Jianqiang |
| author_facet | Xu, Ce Zhao, Jianqiang |
| contents | In this article we shall survey some recent progress on the study of Apéry-like sums which are multiple variable generalizations of the two sums Apéry used in his famous proof of the irrationality of $ζ(2)$ and $ζ(3)$. We only allow the central binomial coefficients to appear in these infinite sums but they can appear either on the numerator or on the denominator. Special values of both types are closely related to the colored multiple zeta values and have played important roles in the calculations of the $\eps$-expansion of multiloop Feynman diagrams. We will summarize several different approaches to computing these sums and prove a few conjectural identities of Z.-W. Sun as corollaries along the way. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_12550 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Apéry-Like Sums and Colored Multiple Zeta Values Xu, Ce Zhao, Jianqiang Number Theory 11M32, 11B65, 33B30 In this article we shall survey some recent progress on the study of Apéry-like sums which are multiple variable generalizations of the two sums Apéry used in his famous proof of the irrationality of $ζ(2)$ and $ζ(3)$. We only allow the central binomial coefficients to appear in these infinite sums but they can appear either on the numerator or on the denominator. Special values of both types are closely related to the colored multiple zeta values and have played important roles in the calculations of the $\eps$-expansion of multiloop Feynman diagrams. We will summarize several different approaches to computing these sums and prove a few conjectural identities of Z.-W. Sun as corollaries along the way. |
| title | Apéry-Like Sums and Colored Multiple Zeta Values |
| topic | Number Theory 11M32, 11B65, 33B30 |
| url | https://arxiv.org/abs/2301.12550 |