Properties of Generalized Freud Polynomials

Abstract

We consider the semi-classical generalized Freud weight function \[w_λ(x;t) = |x|^{2λ+1}\exp(-x^4 +tx^2),\qquad x\in\mathbb{R},\] 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.

26 pages, 8 figures

References in corpus (4)

• A Generalized Freud Weight
• A family of nonlinear difference equations: existence, uniqueness, and asymptotic behavior of positive solutions
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• On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight

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• On Freud-Sobolev type orthogonal polynomials
• Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case II
• On a class of q-orthogonal polynomials and the q-Riemann Hilbert Problem
• Hankel Determinant and Orthogonal Polynomials for a Perturbed Gaussian Weight: from Finite $n$ to Large $n$ Asymptotics
• Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case
• Generalised higher-order Freud weights
• Symmetric Sextic Freud Weight
• On properties of a deformed Freud weight