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arXiv · 0812.2921

On the non-quadraticity of values of the q-exponential function and related q-series

Christian Krattenthaler·Igor Rochev·Keijo Vaananen·Wadim Zudilin
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Abstract

We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;λ)=\sum_{n=0}^\infty\frac{z^n}{\prod_{j=1}^n(q^j-λ)}, \qquad |q|>1, \quad λ\notin q^{\mathbb Z_{>0}}, $$ that includes as special cases the Tschakaloff function ($λ=0$) and the $q$-exponential function ($λ=1$). In particular, we prove the non-quadraticity of the numbers $F_q(α;λ)$ for integral $q$, rational $λ$ and $α\notin-λq^{\mathbb Z_{>0}}$, $α\ne0$.

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Christian Krattenthaler, Igor Rochev, Keijo Vaananen, Wadim Zudilin. 2008-12-15. On the non-quadraticity of values of the q-exponential function and related q-series. https://doi.org/10.4064/aa136-3-4

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