Semi-classical Orthogonal Polynomials Associated with a Modified Gaussian Weight

Abstract

We are concerned with the monic orthogonal polynomials with respect to the modified Gaussian weight

$$\begin{aligned} w(x)=w(x;s):=\textrm{e}^{-N[x^2+s(x^6-x^2)]},\qquad x\in \mathbb {R} \end{aligned}$$

with parameters $N> 0$ and $s\in [0,1]$. Using the ladder operator approach and associated compatibility conditions, we show that the recurrence coefficient $\beta _n(s)$ satisfies a nonlinear fourth-order difference equation, which is the second member of the discrete Painlevé I hierarchy. We find that the orthogonal polynomials satisfy a second-order ordinary differential equation, with all the coefficients expressed in terms of $\beta _n(s)$. By considering the s evolution, we derive the differential-difference equation for the recurrence coefficient $\beta _n(s)$. We also obtain some relations between the Hankel determinant $D_n(s)$, the sub-leading coefficient $\textrm{p}(n,s)$ of the monic orthogonal polynomials and the recurrence coefficient $\beta _n(s)$. Finally, we study the large n asymptotics of the recurrence coefficient $\beta _n(s)$, the sub-leading coefficient $\textrm{p}(n,s)$ and the logarithmic derivative of $D_n(s)$ for fixed $N>0$. We also consider the asymptotics of $\beta _n(s)$ when n/N is fixed as $n\rightarrow \infty$.