Combinatorial multiple Eisenstein series
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866908958198857728 |
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| author | Bachmann, Henrik Burmester, Annika |
| author_facet | Bachmann, Henrik Burmester, Annika |
| contents | We construct a family of $q$-series with rational coefficients satisfying a variant of the extended double shuffle equations, which are a lift of a given $\mathbb{Q}$-valued solution of the extended double shuffle equations. These $q$-series will be called combinatorial (bi-)multiple Eisenstein series, and in depth one they are given by Eisenstein series. The combinatorial multiple Eisenstein series can be seen as an interpolation between the given $\mathbb{Q}$-valued solution of the extended double shuffle equations (as $q\rightarrow 0$) and multiple zeta values (as $q\rightarrow 1$). In particular, they are $q$-analogues of multiple zeta values closely related to modular forms. Their definition is inspired by the Fourier expansion of multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. Our explicit construction is done on the level of their generating series, which we show to be a so-called symmetril and swap invariant bimould. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_17074 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Combinatorial multiple Eisenstein series Bachmann, Henrik Burmester, Annika Number Theory Combinatorics Quantum Algebra 11M32, 11F11 We construct a family of $q$-series with rational coefficients satisfying a variant of the extended double shuffle equations, which are a lift of a given $\mathbb{Q}$-valued solution of the extended double shuffle equations. These $q$-series will be called combinatorial (bi-)multiple Eisenstein series, and in depth one they are given by Eisenstein series. The combinatorial multiple Eisenstein series can be seen as an interpolation between the given $\mathbb{Q}$-valued solution of the extended double shuffle equations (as $q\rightarrow 0$) and multiple zeta values (as $q\rightarrow 1$). In particular, they are $q$-analogues of multiple zeta values closely related to modular forms. Their definition is inspired by the Fourier expansion of multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. Our explicit construction is done on the level of their generating series, which we show to be a so-called symmetril and swap invariant bimould. |
| title | Combinatorial multiple Eisenstein series |
| topic | Number Theory Combinatorics Quantum Algebra 11M32, 11F11 |
| url | https://arxiv.org/abs/2203.17074 |