Finite orthogonal polynomials on a cone

TL;DR

This paper introduces finite orthogonal polynomials on a cone and its surface, deriving their properties and showing their relation to Laguerre polynomials in a limit case.

Contribution

It defines new classes of finite orthogonal polynomials on a cone and surface, along with their differential equations and recurrence relations.

Findings

Derived differential equations for the polynomials

Established recurrence relations for the classes

Connected one class to Laguerre polynomials in a limit

Abstract

The aim of this paper is to study finite orthogonal polynomials on a cone of revolution and its surface. We define two classes of finite orthogonal polynomials on the solid cone and derive their corresponding differential equations and recurrence relations. Furthermore, we demonstrate that, in the limit case, one of these classes reduces to Laguerre polynomials on the cone. Similarly, we establish two families of finite orthogonal polynomials on the surface of the cone and analyze their respective properties.