A note on Dickson polynomials of the third kind and Legendre functions

TL;DR

This paper explores the relationship between Dickson polynomials of the third kind and Legendre functions, showing they satisfy a specific differential equation with solutions involving Legendre functions.

Contribution

It establishes a connection between Dickson polynomials of the third kind and Legendre functions through a differential equation, expanding understanding of their mathematical properties.

Findings

Dickson polynomials of the third kind satisfy a nonhomogeneous second order differential equation.

General solutions involve Legendre functions.

Provides new insights into the differential equations related to these polynomials.

Abstract

In this paper, we show that the Dickson polynomials of the third kind satisfy a nonhomogeneous second order linear ordinary differential equation whose general solution contains Legendre functions.