On the irrationality of a certain multivariate 𝑞 series

Borwein, Peter; Zhou, Ping

doi:10.1090/S0002-9939-03-06941-7

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

Proc. Amer. Math. Soc. 131 (2003), 1989-1998

Abstract:

We prove that for integers $q>1,m\geq 1$ and positive rationals $r_1,r_2,\cdots ,r_m\neq q^j,j=1,2,\cdots ,$ the series \[ \sum _{j=1}^\infty \frac {q^{-j}}{\left ( 1-q^{-j}r_1\right ) \left ( 1-q^{-j}r_2\right ) \cdots \left ( 1-q^{-j}r_m\right ) } \] is irrational. Furthermore, if all the positive rationals $r_1,r_2,\cdots ,r_m$ are less than $q,$ then the series \[ \sum _{j_1,\cdots ,j_m=0}^\infty \frac {r_1^{j_1}\cdots r_m^{j_m}}{ q^{j_1+\cdots +j_m+1}-1} \] is also irrational.

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