On Mittag-Leffler d-orthogonal polynomials

TL;DR

This paper explores the construction and characterization of d-orthogonal polynomials with Hahn's property, focusing on their derivatives and generalizations of Mittag-Leffler polynomials, contributing to the theory of orthogonal polynomial sequences.

Contribution

It introduces a new characterization of Hahn-classical d-orthogonal polynomials via derivatives and constructs specific classes including Laguerre and Mittag-Leffler type polynomials.

Findings

Identification of polynomial solutions as Laguerre d-orthogonal polynomials

Generalization of Mittag-Leffler polynomials

Use of derivative and discrete operators in construction

Abstract

This paper presents a first result of a long term research project dealing with the construction of d-orthogonal polynomials with Hahn's property. We shall show that the latter class could be characterized by expanding a polynomial as a finite sum of first derivatives of the elements of the sequence and we shall explain how this characterization could be used to construct Hahn-classical d-orthogonal polynomials as well. In this paper we look for solutions of linear combinations of the first derivatives of two consecutive elements of the sequence by considering the derivative operator and Delta (discrete) operator. The resulting polynomials constitute a particular class of Laguerre d-orthogonal polynomials and a generalization of Mittag-Leffler polynomials, respectively.