Generalised Airy Polynomials

TL;DR

This paper studies properties of semi-classical orthogonal polynomials related to generalized Airy and sextic Freud weights, focusing on zeros, recurrence coefficients, and their mathematical structure.

Contribution

It introduces new analyses of orthogonal polynomials associated with generalized Airy and sextic Freud weights, including their zeros and recurrence relations.

Findings

Analysis of zeros of the polynomials

Recurrence coefficients characterization

Properties of orthogonal polynomials with generalized weights

Abstract

We consider properties of semi-classical orthogonal polynomials with respect to the generalised Airy weight \[\omega(x;t,\lambda)=x^{\lambda}\exp\left(-\tfrac13x^3+tx\right),\qquad x\in \mathbb{R}^+,\] with parameters $\lambda>-1$ and $t\in \mathbb{R}$. We also investigate the zeros and recurrence coefficients of the polynomials. The generalised sextic Freud weight \[\omega(x;t,\lambda)=|x|^{2\lambda+1}\exp\left(-x^6+tx^2\right), \qquad x\in \mathbb{R},\] arises from a symmetrisation of the generalised Airy weight and we study analogous properties of the polynomials orthogonal with respect to this weight.