Hypergeometric $tau$ Function of the $q$-Painleve Systems of Type $(A_2+A_1)^{(1)}$

We consider a $q$-Painlev'e III equation and a $q$-Painlev'e II equation arising from a birational representation of the affine Weyl group of type $(A_2+A_1)^{(1)}$. We study their hypergeometric solutions on the level of $\tau$ functions.

💡 Research Summary

The paper investigates two discrete Painlevé equations, a $q$‑Painlevé III (q‑P${\mathrm{III}}$) and a $q$‑Painlevé II (q‑P${\mathrm{II}}$), which arise from a birational representation of the affine Weyl group of type $(A_{2}+A_{1})^{(1)}$. The authors first construct the Weyl group generators $s_{0},s_{1},s_{2}$ and the diagram automorphism $\pi$, and they write down their explicit birational actions on the dependent variables $f_{n}$ and $g_{n}$. From these actions the nonlinear $q$‑difference equations are obtained: \