Linear independence of values of Tschakaloff series

In this note we investigate arithmetic properties of values of the Tschakaloff function Tq(z) = Σ^∞_v=0q^-v(v+1)/2_z^v, |q| >1. One of the open problems is proving linear independence of the values of T_q(z) with different q. The only result obtained in this direction, in [1], is very restrictive. We refer the interested reader to the survey [2] for an account of known linear and algebraic independence results for values of Tschakaloff and other q-series