  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
$\alpha\notin-\lambda q^{\mathbb Z_{>0}}$, $\alpha\ne0$.
