An elementary proof of the irrationality of Tschakaloff series

We present a new proof of the irrationality of values of the series
$$ \mathcal T_q(z)=\sum_{n=0}^\infty z^nq^{-n(n-1)/2} $$
in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to $\mathcal T_q(z)$.