The Recurrence Coefficients of Orthogonal Polynomials with a Weight Interpolating between the Laguerre Weight and the Exponential Cubic Weight

Abstract

In this paper, we consider the orthogonal polynomials with respect to the weight w(x)=w(x;s):=xλe−N[x+s(x3−x)],x∈R+, where λ>0, N>0 and 0≤s≤1. By using the ladder operator approach, we obtain a pair of second-order nonlinear difference equations and a pair of differential–difference equations satisfied by the recurrence coefficients αn(s) and βn(s). We also establish the relation between the associated Hankel determinant and the recurrence coefficients. From Dyson’s Coulomb fluid approach, we prove that the recurrence coefficients converge and the limits are derived explicitly when q:=n/N is fixed as n→∞.

Keywords

• orthogonal polynomials
• Laguerre weight
• exponential cubic weight
• ladder operators
• difference equations
• Coulomb fluid