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A study of elliptic gamma function and allies. (English) Zbl 1440.11063
Summary: We study analytic and arithmetic properties of the elliptic gamma function \[ \prod _{m,n=0}^\infty \frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \quad |q|,|p|<1,\] in the regime \(p=q\), in particular, its connection with the elliptic dilogarithm and a formula of S. J. Bloch [Higher regulators, algebraic \(K\)-theory, and zeta functions of elliptic curves. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0958.19001)]. We further extend the results to more general products by linking them to non-holomorphic Eisenstein series and, via some formulae of D. Zagier [Math. Ann. 286, No. 1–3, 613–624 (1990; Zbl 0698.33001)], to elliptic polylogarithms.

MSC:
11F27 Theta series; Weil representation; theta correspondences
11G55 Polylogarithms and relations with \(K\)-theory
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