On symmetries of the nonstationary $\mathrm{P}_\mathrm{II}^{(n)}$ hierarchy and their applications

We study auto-Bäcklund transformations of the nonstationary second Painlevé hierarchy $\mathrm{P}_\mathrm{II}^{(n)}$ depending on $n$ parameters: a parameter $\alpha_n$ and times $t_1, \dots, t_{n-1}$. Using generators $s^{(n)}$ and $r^{(n)}$ of these symmetries, we construct an affine Weyl group $W^{(n)}$ and its extension $\widetilde{W}^{(n)}$ associated with the $n$th member of the hierarchy. We determine rational solutions of $\mathrm{P}_\mathrm{II}^{(n)}$ in terms of Yablonskii–Vorobiev-type polynomials $u_m^{(n)}(z)$. We show that Yablonskii–Vorobiev-type polynomials are related to the polynomial $\tau$-function $\tau_m^{(n)}(z)$ and find their determinant representation in the Jacobi–Trudi form.