# On an extension of generalized coherent pairs of orthogonal polynomials: the classical case

## TL;DR

This paper extends the concept of generalized coherent pairs of orthogonal polynomials, unifying various cases and deriving new recurrence relations and modifications when classical functionals are involved.

## Contribution

It introduces a broader framework for extended coherent pairs, including symmetric cases, and provides explicit formulas for recurrence coefficients and rational modifications.

## Key findings

- Unified extended coherent pairs framework
- Explicit recurrence coefficients for new polynomial systems
- Rational modifications of classical moment functionals

## Abstract

Given two quasi-definite moment functionals, the corresponding orthogonal polynomial systems satisfy an algebraic differential relation(called an extended coherent pair). We study generalizing extended coherent pairs that unify extended coherent pairs and extended symmetric coherent pairs and find the related coefficients. When one of the moment functionals is (strongly) classical, we find another orthogonal polynomial system to find three-term recurrence coefficients. Moreover, we determine the companion moment functional as a rational modification of the classical one.