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Let $q\ge 2$ be an integer. Fourier's proof of irrationality of $e$ adapts to prove the irrationality of $$\Psi_q=\sum_{n\ge0}\frac1{\prod_{k=0}^{n-1}(q^n-q^k)}$$ Is this number knwon to be transcendental?

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  • $\begingroup$ @FedorPetrov From the first sentence I suspect arbitrary integer $q\geq 2$ is allowed. $\endgroup$
    – Wojowu
    Mar 12, 2016 at 16:54
  • $\begingroup$ Indeed, I missed this $\endgroup$ Mar 12, 2016 at 16:55
  • $\begingroup$ Don't know how to proceed from this, but $\Psi_q=G(1/q)$, where $G$ is the Rogers-Ramanujan function. I believe there is a lot known about the cases when are values of $G$ algebraic numbers. $\endgroup$ Mar 12, 2016 at 20:58

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