# On Freud-Sobolev type orthogonal polynomials

## TL;DR

This paper studies Freud-Sobolev orthogonal polynomials, deriving connection formulas, recurrence relations, analyzing zeros and asymptotics, and providing an electrostatic interpretation of their properties.

## Contribution

It introduces new Freud-Sobolev orthogonal polynomials, establishes their connection to Freud polynomials, and explores their zeros, asymptotics, and electrostatic models.

## Key findings

- Derived connection formulas with Freud polynomials
- Established a five-term recurrence relation
- Analyzed zeros and asymptotic behavior

## Abstract

In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner product \begin{equation*} \left\langle p,q\right\rangle _{s}=\int_{\mathbb{R}}p(x)q(x)e^{-x^{4}}dx+M_{0}p(0)q(0)+M_{1}p^{\prime }(0)q^{\prime }(0), \end{equation*}% where $p,q$ are polynomials, $M_{0}$ and $M_{1}$ are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as a consequence, a five term recurrence relation for such polynomials is obtained. The location of their zeros as well as their asymptotic behavior is studied. Finally, an electrostatic interpretation of them in terms of a logarithmic interaction in the presence of an external field is given.