Some identities for quasimodular functions

For the first time modular functions for the proof of transcendence of numbers were used in 1935 in joint article of K. Mahler and Y. Popken. They proved that at any complex number $\tau$, $\Im\tau>0$, the set $E_{2}(\tau)$, $E_{4}(\tau)$, $E_{6}(\tau)$ (Eisenstein series) contains at least one transcendental number. The first transcendence result about values of the modular invariant $j(\tau)$ has been proved in 1937 by Th. Schneider. For the proof he used properties of Weierstrass's elliptic functions, but this way seemed to him unnatural and Schneider formulated a problem to find the modular proof of his theorem. Then Mahler formulated a hypothesis about transcendence at any $\tau$, $\Im\tau>0$, at least one of two numbers $e^{ 2\pi i\tau }$ and $j (\tau) $. Now a modular proof of Schneider's theorem is still not found. It is also open the complete hypothesis of Mahler -Manin about values of modular invariant and exponential function $a^{\tau}$ for algebraic $a\ne 0,\, $1. In the talk we will discuss some results in this area and some attempts to use others modular and quasimodular functions, about further advances in this area.