# Characterization of quadratic Cauchy-Stieltjes Kernel families by orthogonality of polynomials

## TL;DR

This paper characterizes quadratic Cauchy-Stieltjes Kernel families through the orthogonality properties of associated polynomial sequences, providing a new mathematical insight into their structure.

## Contribution

It establishes that the quadratic variance function uniquely corresponds to the orthogonality of the related polynomial sequence within CSK families.

## Key findings

- Quadratic variance functions are characterized by polynomial orthogonality.
- Orthogonality of polynomials uniquely identifies quadratic CSK families.
- Provides a mathematical link between variance functions and polynomial properties.

## Abstract

In this paper we specify some facts about the sequence of polynomials associated to a \CSK family and we prove that quadratic variance function is characterized by the property of orthogonality of these polynomials.