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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.

##### MSC:
 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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