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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
##### Citations:
Zbl 0958.19001; Zbl 0698.33001
Full Text:
##### References:
 [1] Bloch, S.J.: Higher regulators, algebraic $$K$$-theory, and zeta functions of elliptic curves. In: Lecture Notes (UC Irvine, 1977); CRM Monograph Series, vol. 11. American Mathematical Society, Providence (2000) [2] Duke, William; Imamoḡlu, Özlem, On a formula of Bloch, Functiones et Approximatio Commentarii Mathematici, 37, 109-117, (2007) · Zbl 1213.11141 [3] Felder, G.; Varchenko, A., The elliptic gamma function and \(\text{ SL }(3,{\mathbb{Z}})
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