An identity for generating series of deformations of multiple zeta values within an algebraic framework
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911049079324672 |
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| author | Takeyama, Yoshihiro |
| author_facet | Takeyama, Yoshihiro |
| contents | Bachmann proves an identity expressing the generating series of MacMahon's generalized sum-of-divisors $q$-series in terms of Eisenstein series. MacMahon's $q$-series can be regarded as a $q$-analogue of the multiple zeta value $ζ(2, 2, \ldots , 2)$, up to a power of $1-q$. Based on this observation, we generalize Bachmann's identity within an algebraic framework and prove a general identity. As a byproduct, we obtain a formula for the generating series of another deformation of multiple zeta values defined by the author. In this formula, periodlike functions introduced by Lewis and Zagier appear as a counterpart of Eisenstein series. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_16131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An identity for generating series of deformations of multiple zeta values within an algebraic framework Takeyama, Yoshihiro Number Theory Quantum Algebra 11M32, 33E20, 05A30 Bachmann proves an identity expressing the generating series of MacMahon's generalized sum-of-divisors $q$-series in terms of Eisenstein series. MacMahon's $q$-series can be regarded as a $q$-analogue of the multiple zeta value $ζ(2, 2, \ldots , 2)$, up to a power of $1-q$. Based on this observation, we generalize Bachmann's identity within an algebraic framework and prove a general identity. As a byproduct, we obtain a formula for the generating series of another deformation of multiple zeta values defined by the author. In this formula, periodlike functions introduced by Lewis and Zagier appear as a counterpart of Eisenstein series. |
| title | An identity for generating series of deformations of multiple zeta values within an algebraic framework |
| topic | Number Theory Quantum Algebra 11M32, 33E20, 05A30 |
| url | https://arxiv.org/abs/2506.16131 |