On properties of a deformed Freud weight

TL;DR

This paper investigates the recurrence coefficients of polynomials orthogonal with respect to a deformed Freud weight, revealing their connection to Painlevé equations, asymptotic behaviors, and the associated differential equations, including the Heun equation.

Contribution

It establishes the discrete Painlevé I equation for the recurrence coefficients and derives a second order linear ODE, including asymptotic analysis and the Hankel determinant behavior.

Findings

Recurrence coefficients satisfy discrete Painlevé I and differential equations.

Asymptotic behaviors are characterized for different parameter regimes.

The associated linear ODE approaches a biconfluent Heun equation as n→∞.

Abstract

We study the recurrence coefficients of the monic polynomials $P_n(z)$ orthogonal with respect to the deformed (also called semi-classical) Freud weight \begin{equation*} w_{\alpha}(x;s,N)=|x|^{\alpha}{\rm e}^{-N\left[x^{2}+s\left(x^{4}-x^{2}\right)\right]}, ~~x\in\mathbb{R}, \end{equation*} with parameters $\alpha>-1,~N>0,~s\in[0,1]$. We show that the recurrence coefficients $\beta_{n}(s)$ satisfy the first discrete Painlev\'{e} equation (denoted by d${\rm P_{I}}$), a differential-difference equation and a second order nolinear ordinary differential equation (ODE) in $s$. Here $n$ is the order of the Hankel matrix generated by $w_{\alpha}(x;s,N)$. We describe the asymptotic behavior of the recurrence coefficients in three situations, (i) $s\rightarrow0$, $n,N$ finite, (ii) $n\rightarrow\infty$, $N$ finite, (iii) $n, N\rightarrow\infty$, such that the radio $r:=\frac{n}{N}$ is bounded away from $0$ and closed to $1$. We also investigate the existence and uniqueness for the positive solutions of the d${\rm P_{I}}$. Further more, we derive, using the ladder approach, a second order linear ODE satisfied by the polynomials $P_n(z)$. It is found as $n\rightarrow\infty$, the linear ODE turns to be a biconfluent Heun equation. This paper concludes with the study of the Hankel determinant, $D_{n}(s)$, associated with $w_{\alpha}(x;s,N)$ when $n$ tends to infinity.