Integrated Pearson family and orthogonality of the Rodrigues polynomials: A review including new results and an alternative classification of the Pearson system

TL;DR

This paper reviews the Integrated Pearson family, its relation to orthogonal Rodrigues polynomials, introduces new results, and provides an algorithm to classify Pearson densities within this family.

Contribution

It offers a new classification of the Pearson system based on polynomial orthogonality and presents an algorithm for identifying densities in the Integrated Pearson Family.

Findings

The Integrated Pearson Family is a subset of the Pearson system characterized by polynomial orthogonality.

New properties and recurrence relations for Rodrigues polynomials are established.

An algorithm to determine if a Pearson density belongs to the Integrated Pearson Family is proposed.

Abstract

An alternative classification of the Pearson family of probability densities is related to the orthogonality of the corresponding Rodrigues polynomials. This leads to a subset of the ordinary Pearson system, the Integrated Pearson Family. Basic properties of this family are discussed and reviewed, and some new results are presented. A detailed comparison between the integrated Pearson family and the ordinary Pearson system is presented, including an algorithm that enables to decide whether a given Pearson density belongs to the integrated system, or not. Recurrences between the derivatives of the corresponding orthonormal polynomial systems are also given.