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Elliptic polylogarithms: An analytic theory. (English) Zbl 0905.11028
The starting point is the \(n\)-th Debye polylogarithm, which is the multivalued function \[ \Lambda_n (\xi)= \int^\infty_\xi {t^{n-1} \over (n-1)!} {dt\over \exp (-2\pi it) -1}. \] This is single-valued in the upper half plane if the path of integration is chosen to be the vertical line from \(\xi\) to \(i\infty\). The branch \(\Lambda^+_n (\xi)\) is the continuation to the whole plane without \((-\infty,0) \cup (1,\infty)\). The Debye elliptic polylogarithm \(\Lambda_{m,n} (\xi, \tau)\) on \(\mathbb{C} \times H\) is defined by regularisation of the divergent series \(\sum^\infty_{j=- \infty} j^m\Lambda^+_n (\xi+ j\tau)\). (Compare the definition \(\zeta(-m)= \sum^\infty_{j=1} j^m\).) The main result is that the generating function \(\Lambda (\xi,\tau; X,Y)= \sum_{m\geq 0, n \geq 1} \Lambda_{m,n} (\xi,\tau) (-Y)^{n-1} X^m\) has modular properties and can be expressed in terms of non-holomorphic Eisenstein series.

11G05 Elliptic curves over global fields
11F11 Holomorphic modular forms of integral weight
33B10 Exponential and trigonometric functions
33B15 Gamma, beta and polygamma functions
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