Properties of Generalized Freud Polynomials

TL;DR

This paper studies the asymptotic properties, recurrence coefficients, and zeros of generalized Freud orthogonal polynomials associated with a semi-classical weight function, extending understanding of their behavior for large degrees and parameters.

Contribution

It provides new asymptotic analysis, existence and uniqueness results, and zero distribution properties for generalized Freud polynomials with a semi-classical weight.

Findings

Asymptotic behavior of orthogonal polynomials analyzed

Recurrence coefficients' limits and properties established

Zeros of the polynomials characterized

Abstract

We consider the semi-classical generalized Freud weight function \[w_{\lambda}(x;t) = |x|^{2\lambda+1}\exp(-x^4 +tx^2),\qquad x\in\mathbb{R},\] with $ \lambda>-1$ and $t\in\mathbb{R}$ parameters. We analyze the asymptotic behavior of the sequences of monic polynomials that are orthogonal with respect to $w_{\lambda}(x;t)$, as well as the asymptotic behavior of the recurrence coefficient, when the degree, or alternatively, the parameter $t$, tend to infinity. We also investigate existence and uniqueness of positive solutions of the nonlinear difference equation satisfied by the recurrence coefficients and prove properties of the zeros of the generalized Freud polynomials.