On a new class of 2-orthogonal polynomials, II_The integral representations

2022/12/22 by Khalfa Douak, Douak, Khalfa, P. Maroni +1 · 1 citation
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2212.11949

openalex publication_date 2022/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A new class of 2-orthogonal polynomials satisfying orthogonality conditions with respect to a pair of linear functionals (u₀,u₁) was presented in Douak K & Maroni P [On a new class of 2-orthogonal polynomials, I: the recurrence relations and some properties. Integral Transforms Spec Funct. 2021;32(2):134-153]. Six interesting special cases were pointed out there. For each case, we precisely deal with the integral representation problem for the functionals associated to these polynomials. The focus is on the matrix differential equation (\bfΦU)'+\bfΨU=0, with \bf U=^t(u₀ , u₁) and \bfΦ, \bfΨ are 2×2 polynomial matrices, from which we establish the differential equations satisfied by the two functionals. Based on this, depending on the case, we show that u₀ and u₁ are represented via weight functions supported on the real line or positive real line and defined in terms of various special functions. In order for certain integral representations to exist, addition of Dirac mass is necessary.

Cited by

• Three-term Recurrence Relation with Arbitrary Degree Steps for Orthogonal Polynomials

Related

• Characterization of classical orthogonal polynomials in two variables
• Matrix orthogonal polynomials whose derivatives are also orthogonal
• Explicit construction of matrix-valued orthogonal polynomials of arbitrary size
• Integral and hypergeometric representations for multiple orthogonal polynomials
• Second order differential operators having several families of orthogonal matrix polynomials as eigenfunctions