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

TL;DR

This paper investigates the orthogonality properties of Dickson polynomials of the (k+1)-th kind over complex numbers, establishing their orthogonality, deriving an integral representation, and computing moments.

Contribution

It introduces the orthogonality of Dickson polynomials of the (k+1)-th kind and provides explicit formulas for their moments and integral representation.

Findings

Dickson polynomials of the (k+1)-th kind are orthogonal with respect to a specific moment functional.

An explicit integral representation for the moment functional is derived.

Explicit moments of the functional are computed.

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 L_{k}. We find an integral representation for L_{k} and compute explicit expressions for all of its moments.