$q$-Hypergeometric Functions and Irrationality Measures

In this talk I shall present a $q$-analogue of the Rhin-Viola method for the analysis of $\Phi$-adic valuations of the $q$-gamma factors that occur in the basic Euler-Pochhammer integral representation of the Heine series $_2\phi_1$(a $q$-analogue of the Gauss hypergeometric function $_2F_1$). We shall also discuss its implications to the diophantine approximation of certain $q$-hypergeometric function. Namely, this approach yields the best known irrationality measures for
$$
\log_q(1-z) = \sum_{n=1}^{\infty}\frac{zq^n}{1-zq^n},\quad |z|\leq 1,\quad q = 1/p,\quad p\in \mathbb{Z}\setminus\{0,\pm 1\},
$$
a $q$-analogue of the ordinary logarithmic function, and in particular for
$$
\log_q 2 = \sum_{n=1}^{\infty}\frac{(-1)^nq^n}{1-q^n},\quad \textrm{and} \quad \zeta_q(1) = \sum_{n=1}^{\infty}\frac{q^n}{1-q^n},
$$
a $q$-analogue of $\log 2$ and the $q$-harmonic series.