Apéry-type approximations and irrationality measures for certain $q$-series

Abstract

We construct a three-parameter family of rational approximations to values of $q$-hypergeometric series. Using these approximations, we prove that, for every integer $x$ with $|x|\ge 2$, the values at $r=x^{-1}$ of Ramanujan's theta function $\psi (r)={\sum }_{n\ge 0}{r}^{n(n+1)/2}$, the generating function $\Delta (r)={\sum }_{m\ge 0}d(2m+1)r^m$ of the divisor function restricted to odd integers, and the generating function $B_4(r)={\sum }_{n\ge 0}{b}_4(n)r^n$ for $4$-regular partitions are irrational. We further obtain the upper bounds $18/7$, $18{\pi }^2/(7{\pi }^2-24)$, and $3$, respectively, for their irrationality measures. We also show that one of the constructed approximations coincides with the Padé approximation to a Lambert series due to Coussement--Smet.

arXiv comment37 pages. Comments welcome!