On Freud-Sobolev type orthogonal polynomials

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.

arXiv commentthree figures

References in corpus (6)

• The recurrence coefficients of semi-classical Laguerre polynomials and the fourth Painlevé equation
• Painleve I asymptotics for orthogonal polynomials with respect to a varying quartic weight
• A Generalized Freud Weight
• Properties of Generalized Freud Polynomials
• On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight
• Sobolev Freud polynomials

Cited by in corpus (2)

• On certain properties of perturbed Freud-type weight: a revisit
• Ladder operators and a second--order difference equation for general discrete Sobolev orthogonal polynomials