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

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

2008-12-15

arxiv.org/abs/0812.2921v1

Acta Arith. 136 (2009), no. 3, 243--269

Mathematics

Number Theory

27 pages

Scientific paper

10.4064/aa136-3-4

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

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