A study of elliptic gamma function and allies

We study analytic and arithmetic properties of the elliptic gamma function $$\prod _{m,n=0}^{\mathrm{\infty }}\frac{1-x^{-1}{q}^{m+1}{p}^{n+1}}{1-x{q}^m{p}^n},\mathrm{\mid }q\mathrm{\mid },\mathrm{\mid }p\mathrm{\mid }<1,$$ in the regime $p=q$; 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 Eistenstein series and, via some formulae of D. Zagier, to elliptic polylogarithms.