A generic classification of exceptional orthogonal X1-polynomials based on Pearson distributions family

TL;DR

This paper provides a comprehensive classification of exceptional orthogonal X1-polynomials using Pearson distributions, introducing six differential equations and analyzing their polynomial solutions in detail.

Contribution

It offers a new, unified framework for classifying exceptional orthogonal polynomials based on Pearson distributions, expanding understanding of their differential equations.

Findings

Six special differential equations identified

Polynomial solutions analyzed in detail

Classification framework for exceptional orthogonal polynomials established

Abstract

The so-called exceptional orthogonal X1-polynomials arise as eigen functions of a Sturm-Liouville problem. In this paper, a generic classification of these polynomials is presented based on Pearson distributions family. Then, six special differential equations of the aforesaid classification are introduced and their polynomial solutions are studied in detail.