Differential and Difference Equations for Recurrence Coefficients of Orthogonal Polynomials with a Singularly Perturbed Laguerre-type Weight

TL;DR

This paper derives nonlinear difference and differential-difference equations for recurrence coefficients of orthogonal polynomials with a singularly perturbed Laguerre-type weight, and analyzes their asymptotic behavior.

Contribution

It introduces a new system of nonlinear difference equations and differential-difference equations for these recurrence coefficients, advancing understanding of their asymptotics.

Findings

Derived nonlinear second-order difference equations for recurrence coefficients

Established large n asymptotic expansions of the coefficients

Obtained a system of differential-difference equations for the coefficients

Abstract

We are concerned with the monic orthogonal polynomials with respect to a singularly perturbed Laguerre-type weight. By using the ladder operator approach, we derive a complicated system of nonlinear second-order difference equations satisfied by the recurrence coefficients. This allows us to derive the large $n$ asymptotic expansions of the recurrence coefficients. In addition, we also obtain a system of differential-difference equations for the recurrence coefficients.