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 \[w(x;T_{m};\lambda) = |x|^{2\lambda+1}\exp\biggl(-\sum_{k=1}^m t_k x^{2k}\biggr),\quad m \in \mathbb{Z}^+,\; t_{k} \in \mathbb{R} , x\in\mathbb{R}\setminus\{0\},\] where \(T_{m}=\{t_{1},t_{2},\cdots, t_{m}\}\), $t_{m}>0$ and \(\lambda>-1\).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 $n\to\infty$, 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 $w(x;t_1,t_2;\lambda)=|x|^{2\lambda+1} \exp\bigl(-x^{10}-t_2x^4-t_1x^2\bigr)$.