Orthogonality of the Dickson polynomials of the $(k+1)$-th kind

Diego Dominici

doi:10.33044/revuma.v62n1a01

Abstract

We study the Dickson polynomials of the (k + 1)-th kind over the field of complex numbers. We show that they are a family of co-recursive orthogonal polynomials with respect to a quasi-definite moment functional Lk. We find an integral representation for Lk and compute explicit expressions for all of its moments.

Highlights

Let n ∈ N, Fq be a finite field, and a ∈ Fq
In [58], Wang and Yucas extended the Dickson polynomials to a family depending on a new parameter k ∈ N0, which they called Dickson polynomials of the (k + 1)-th kind
Our motivation is the three-term recurrence relation (1.3), which suggests that the Dickson polynomials of the (k + 1)-th kind form a family of orthogonal polynomials with respect to some linear functional Lk

Summary

Introduction

Let n ∈ N, Fq be a finite field, and a ∈ Fq. The Dickson polynomials Dn(x; a), defined by (see [29, 9.6.1]). In [58], Wang and Yucas extended the Dickson polynomials to a family depending on a new parameter k ∈ N0, which they called Dickson polynomials of the (k + 1)-th kind Our motivation is the three-term recurrence relation (1.3), which suggests that the Dickson polynomials of the (k + 1)-th kind form a family of orthogonal polynomials with respect to some linear functional Lk. from (1.4) we see that for k > 2 the polynomials Dn,k(x; a) may have a pair of purely imaginary roots. The article is organized as follows: in Section 2, we derive some of the main properties of the Dickson polynomials of the (k + 1)-th kind, including different expressions, a hypergeometric representation, differential equations, and a generating function. In the hope that our results will be of interest to researchers outside the field of orthogonal polynomials and special functions, we have made the paper as self-contained as possible

Properties of Dickson polynomials

Orthogonal polynomials

Main results