An elementary proof of the irrationality of Tschakaloff series

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