On the irrationality of a certain 𝑞 series

Borwein, Peter; Zhou, Ping

doi:10.1090/S0002-9939-99-04722-X

On the irrationality of a certain $q$ series
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by Peter B. Borwein and Ping Zhou PDF

Proc. Amer. Math. Soc. 127 (1999), 1605-1613

Erratum: Proc. Amer. Math. Soc. 132 (2004), 3131-3131.

Abstract:

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

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