Generalized Freud weight, discrete Painlevé I hierarchy and full asymptotics of Hankel determinants

Wen-Gao Long; Lin-Rong Tang; Chao Min

paper

Generalized Freud weight, discrete Painlevé I hierarchy and full asymptotics of Hankel determinants

arXiv:2608.08455

Abstract

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 $ln β_{n}$ 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 $p (n; T_{m}; λ)$ , 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 $w (x; t_{1}, t_{2}; λ) = ∣ x ∣^{2 λ + 1} exp (- x^{10} - t_{2} x^{4} - t_{1} x^{2})$ .