Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finite orthogonal polynomials on a cone

Ömer Faruk Et, Esra Çekirdek, Rabia Aktaş Karaman

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.

Finite orthogonal polynomials on a cone

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.
Paper Structure (17 sections, 12 theorems, 143 equations)

This paper contains 17 sections, 12 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Finite orthogonal polynomials of one variable
  4. First Class of Finite OPs:
  5. Second Class of Finite OPs:
  6. Spherical Harmonics
  7. OPs on the unit ball
  8. OPs in and on a cone
  9. OPs on the solid cone
  10. OPs on the surface of the cone
  11. Finite orthogonal polynomials in and on a cone
  12. Finite orthogonal polynomials on a solid cone
  13. First finite class on the cone
  14. Second finite class on the cone
  15. Finite orthogonal polynomials on the surface of a cone
  16. ...and 2 more sections

Key Result

Proposition 3.1

Let ${\mathbb{P}}_{m}=\{P_{\mathbf{k}}:|{\mathbf{k}}|=m\}$ be an orthonormal basis of ${\mathcal{V}}_{m}^{d}({\mathbb{B}}^{d},{\mathsf{w}}_{\mu })$ for $m\leq n$ . Let $\alpha =\mu +\frac{d-1}{2}.$ Define Then $\left \{ M_{\mathbf{k,}n}^{\left( p,q,\mu \right) }\left( x,t\right) :\left \vert \mathbf{k}\right \vert =m,~\,0\leq m\leq n\right \}$ is an orthogonal basis of ${\mathcal{V}}_{n}^{d}(\math

Theorems & Definitions (26)

  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 1
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 2
  • ...and 16 more