On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight

TL;DR

This paper studies polynomials orthogonal with respect to a Freud-like weight modified by an Uvarov term, analyzing their properties, zero distribution, and dynamic behavior as the parameter varies.

Contribution

It introduces and analyzes a new class of orthogonal polynomials with a modified Freud weight, including their ladder operators, recurrence relations, and electrostatic zero distribution.

Findings

Derived ladder operators and holonomic equations for the polynomials.

Provided an electrostatic interpretation of the zeros' distribution.

Established a nonlinear difference equation for recurrence coefficients.

Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potential $\left\langle p,q\right\rangle_{M}=\int_{\mathbb{R}}p(x)q(x)e^{-x^{4}+2tx^{2}}dx+Mp(0)q(0).$ We analyze some properties of these polynomials, such as the ladder operators and the holonomic equation that they satisfy and, as an application, we give an electrostatic interpretation of their zero distribution in terms of a logarithmic potential interaction under the action of an external field. It is also shown that the coefficients of their three term recurrence relation satisfy a nonlinear difference string equation. Finally, an equation of motion for their zeros in terms of their dependence on $t$ is given.