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

We investigate arithmetic properties of values of the entire function F(z) = F_q(z;λ)=[formula could not be replicated], |q|>1, λ ∉ q^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<sub>q/sub>(α;λ) for integral q, rational λ and α ∉ -λq^ℤ>0, α ≠ 0