Generalized Freud weight, discrete Painlevé I hierarchy and full asymptotics of Hankel determinants
In this paper, we investigate the monic orthogonal polynomials $P_{n}(x;T_{m};λ)$ and the Hankel determinants $D_{n}(T_{m}; λ)$ associated with the generalized Freud weight \[w(x;T_{m};λ) = |x|^{2λ+1}\exp\biggl(-\sum_{k=1}^m t_k x^{2k}\biggr),\quad m \in \mathbb{Z}^+,\; t_{k} \in \mathbb{R} , x\in\mathbb{R}\setminus\{0\},\] where \(T_{m}=\{t_{1},t_{2},\cdots, t_{m}\}\), $t_{m}>0$ and \(λ>-1\).By employing ladder operators and compatibility conditions, we find that all members of the discrete Painlevé I hierarchy have a unified structure and the recurrence coefficient \(β_n\) of $P_{n}(x;T_{m};λ)$ satisfies the $m$-th member of the discrete Painlevé I hierarchy. Besides, we derive the second-order differential equation satisfied by $P_{n}(x;T_{m};λ)$, the partial derivatives of the recurrence coefficients \(β_n\) with respect to parameters \(t_1, t_2, \dots, t_{m-1}\) and the corresponding differential identities for $D_{n}(T_{m}; λ)$. Based on the discrete Painlevé I hierarchy and the above differential identities, we obtain new partial differential equations satisfied by $\lnβ_n$ and $\ln D_{n}(T_{m}; λ)$.Using the discrete Painlevé I hierarchy and the asymptotic theory of linear difference equations, we derive the full asymptotic expansions of the recurrence coefficient $β_n$, the nontrivial leading coefficient $\mathrm{p}(n; T_m; λ)$, and the Hankel determinant $D_n(T_m; λ)$ as $n\to\infty$, for general $T_m$ and $λ>-1$. Notably, while the logarithmic term $\ln n$ appears in the leading-order contributions, it is absent from the remainder terms in these expansions.We illustrate our results under the specific decic Freud weight $w(x;t_1,t_2;λ)=|x|^{2λ+1} \exp\bigl(-x^{10}-t_2x^4-t_1x^2\bigr)$.