AI 中文总结
本文研究广义Freud权对应的正交多项式与Hankel行列式，揭示离散Painlevé I hierarchy的统一结构，推导相关微分方程与恒等式，得到这些量的完全渐近展开并以十次Freud权为例说明。
AI 中文摘要
本文研究与广义Freud权相关的首一正交多项式$P_{n}(x;T_{m};\boldsymbol{\u03bb})$及Hankel行列式$D_{n}(T_{m}; \boldsymbol{\u03bb})$，其中广义Freud权为$w(x;T_{m};\boldsymbol{\u03bb}) = |x|^{2\boldsymbol{\u03bb}+1}\boldsymbol{\u00b7}\boldsymbol{\u03bexp}\biggl(-\boldsymbol{\u03a3}_{k=1}^m t_k x^{2k}\biggr)$，$m$为正整数，$t_k$为实数，$x$属于非零实数集，$T_{m}=\boldsymbol{\u007bt}_{1},t_{2},\boldsymbol{\u00b7}\boldsymbol{\u00b7}\boldsymbol{\u00b7}, t_{m}\boldsymbol{\u007d}$，$t_{m}>0$，$\boldsymbol{\u03bb}>-1$。通过运用 ladder算子与相容性条件，发现离散Painlevé I hierarchy的所有成员具有统一结构，且$P_{n}(x;T_{m};\boldsymbol{\u03bb})$的递推系数$\beta_n$满足离散Painlevé I hierarchy的第$m$个成员关系。此外，推导得到$P_{n}(x;T_{m};\boldsymbol{\u03bb})$满足的二阶微分方程、递推系数$\beta_n$关于参数$t_1,t_2,\boldsymbol{\u00b7}\boldsymbol{\u00b7}\boldsymbol{\u00b7}, t_{m-1}$的偏导数，以及$D_{n}(T_{m}; \boldsymbol{\u03bb})$对应的微分恒等式。基于离散Painlevé I hierarchy与上述微分恒等式，得到$\boldsymbol{\u03b2}_n$和$\boldsymbol{\u03b9}\boldsymbol{\u03bd}D_{n}(T_{m}; \boldsymbol{\u03bb})$满足的新偏微分方程。利用离散Painlevé I hierarchy与线性差分方程的渐近理论，针对一般$T_m$和$\boldsymbol{\u03bb}>-1$，推导得到当$n\to\boldsymbol{\u221e}$时，递推系数$\beta_n$、非零首项系数$\boldsymbol{\u0070}(n; T_m; \boldsymbol{\u03bb})$及Hankel行列式$D_n(T_m; \boldsymbol{\u03bb})$的完全渐近展开。值得注意的是，对数项$\boldsymbol{\u03b9}\boldsymbol{\u03bd}n$出现在首阶贡献中，但在余项中不存在。以特定的十次Freud权$w(x;t_1,t_2;\boldsymbol{\u03bb})=|x|^{2\boldsymbol{\u03bb}+1} \boldsymbol{\u03bexp}\bigl(-x^{10}-t_2x^4-t_1x^2\bigr)$为例说明所得结果。
英文摘要
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 $w(x;t_1,t_2;λ)=|x|^{2λ+1} \exp\bigl(-x^{10}-t_2x^4-t_1x^2\bigr)$.
