Generalized Freud weight, discrete Painlev\'{e} I hierarchy and full asymptotics of Hankel determinants

In this paper, we investigate the monic orthogonal polynomials $P_n(x;T_m;\lambda )$ and the Hankel determinants $D_n(T_m;\lambda )$ associated with the generalized Freud weight $$w(x;T_m;\lambda )=\mathrm{\mid }x{\mathrm{\mid }}^{2\lambda +1}\exp (-\sum _{k=1}^m{t}_k{x}^{2k}),m\in {\mathbb{Z}}^{+},t_k\in \mathbb{R},x\in \mathbb{R}\setminus \{0\},$$ where $T_m=\{t_1,t_2,\cdots ,t_m\}$, $t_m>0$ and $\lambda >-1$.By employing ladder operators and compatibility conditions, we find that all members of the discrete Painlev'{e} I hierarchy have a unified structure and the recurrence coefficient ${\beta }_n$ of $P_n(x;T_m;\lambda )$ satisfies the $m$-th member of the discrete Painlev'{e} I hierarchy. Besides, we derive the second-order differential equation satisfied by $P_n(x;T_m;\lambda )$, the partial derivatives of the recurrence coefficients ${\beta }_n$ with respect to parameters $t_1,t_2,\dots ,t_{m-1}$ and the corresponding differential identities for $D_n(T_m;\lambda )$. Based on the discrete Painlev'{e} I hierarchy and the above differential identities, we obtain new partial differential equations satisfied by $\ln {\beta }_n$ and $\ln D_n(T_m;\lambda )$.Using the discrete Painlev'{e} I hierarchy and the asymptotic theory of linear difference equations, we derive the full asymptotic expansions of the recurrence coefficient ${\beta }_n$, the nontrivial leading coefficient $\mathrm{p}(n;T_m;\lambda )$, and the Hankel determinant $D_n(T_m;\lambda )$ as $n\to \mathrm{\infty }$, for general $T_m$ and $\lambda >-1$. 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 $w(x;t_1,t_2;\lambda )=\mathrm{\mid }x{\mathrm{\mid }}^{2\lambda +1}\exp (-x^{10}-t_2{x}^4-t_1{x}^2)$.