A study of elliptic gamma function and allies

We study analytic and arithmetic properties of the elliptic gamma function $$\begin{aligned} \prod _{m,n=0}^\infty \frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \quad |q|,|p|<1, \end{aligned}$$ ∏ m , n = 0 ∞ 1 - x - 1 q m + 1 p n + 1 1 - x q m p n , | q | , | p | < 1 , in the regime $$p=q$$ p = q , in particular, its connection with the elliptic dilogarithm and a formula of S. Bloch. We further extend the results to more general products by linking them to non-holomorphic Eisenstein series and, via some formulae of D. Zagier, to elliptic polylogarithms.