  In this work, we give a purely analytic introduction to the phenomenon of
mirror symmetry for quintic threefolds via classical hypergeometric functions
and differential equations for them. Starting with a modular map and recent
transcendence results for its values, we regard a mirror map $z(q)$ as a
concept generalizing the modular one. We give an alternative approach
demonstrating the existence of non-linear differential equations for the mirror
map, and exploit both an elegant construction of Klemm-Lian-Roan-Yau and the Ax
theorem to prove that the Yukawa coupling $K(q)$ does not satisfy any algebraic
differential equation of order less than 7 with coefficients from
$\mathbb{C}(q)$.
