On Semi-Classical Orthogonal Polynomials Associated with a Modified Sextic Freud-Type Weight

Abey S. Kelil and Appanah R. Appadu
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Abey S. Kelil: Department of Mathematics and Applied Mathematics, Nelson Mandela University, Port Elizabeth 6019, South Africa
Appanah R. Appadu: Department of Mathematics and Applied Mathematics, Nelson Mandela University, Port Elizabeth 6019, South Africa

Mathematics, 2020, vol. 8, issue 8, 1-28

Abstract: Polynomials that are orthogonal with respect to a perturbation of the Freud weight function by some parameter, known to be modified Freudian orthogonal polynomials, are considered. In this contribution, we investigate certain properties of semi-classical modified Freud-type polynomials in which their corresponding semi-classical weight function is a more general deformation of the classical scaled sextic Freud weight | x | α exp ( − c x 6 ) , c > 0 , α > − 1 . Certain characterizing properties of these polynomials such as moments, recurrence coefficients, holonomic equations that they satisfy, and certain non-linear differential-recurrence equations satisfied by the recurrence coefficients, using compatibility conditions for ladder operators for these orthogonal polynomials, are investigated. Differential-difference equations were also obtained via Shohat’s quasi-orthogonality approach and also second-order linear ODEs (with rational coefficients) satisfied by these polynomials. Modified Freudian polynomials can also be obtained via Chihara’s symmetrization process from the generalized Airy-type polynomials. The obtained linear differential equation plays an essential role in the electrostatic interpretation for the distribution of zeros of the corresponding Freudian polynomials.

Keywords: semi-classical orthogonal polynomials; Freud-type weights; moments; recurrence coefficients; difference equations; differential equations; zeros (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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