General modular quantum dilogarithm and beta-integrals

The modular quantum dilogarithm was suggested by Faddeev in 1994 (in the theory of special functions it was called the hyperbolic gamma function). It is built with the help of the simplest modular transformation from the group SL(2,Z) applied to the q-Pochhammer symbol. Recently a generalization of this function was suggested, which uses an arbitrary transformation from SL(2,Z). In the talk I will describe this function and present a proof of the evaluation formula for a general univariate hyperbolic beta-integral built with its help.