On the irrationality of a certain 𝑞 series

Abstract

We prove that if q q is an integer greater than one, r r and s s are any positive rationals such that 1 + q m r − q 2 m s ≠ 0 1+q^mr-q^{2m}s\neq 0 for all integers m ≥ 0 m\geq 0 , then \[ ∑ j = 0 ∞ 1 1 + q j r − q 2 j s \sum _{j=0}^\infty \frac 1{1+q^jr-q^{2j}s} \] is irrational and is not a Liouville number.