# On another extension of coherent pairs of measures

## TL;DR

This paper investigates the structure of orthogonal polynomial sequences related by derivatives and polynomial modifications, establishing conditions under which they are semiclassical and introducing the concept of -coherent pairs with specific indices and orders.

## Contribution

It extends the theory of coherent pairs by characterizing when such polynomial sequences are semiclassical and defining a new class of -coherent pairs with rational modifications.

## Key findings

- Orthogonal polynomial sequences are semiclassical under certain derivative relations.
- Moment functionals are related via rational modifications.
- Introduction of -coherent pairs with specific indices and orders.

## Abstract

Let $M$ and $N$ be fixed non-negative integer numbers and let $\pi_N$ be a polynomial of degree $N$. Suppose that $(P_n)_{n\geq0}$ and $(Q_n)_{n\geq0}$ are two orthogonal polynomial sequences such that %their derivatives of orders $k$ and $m$ (respectively) satisfy the structure relation $$ \pi_N(x)\,P_{n+m}^{(m)}(x)= \sum_{j=n-M}^{n+N}r_{n,j}Q_{j+k}^{(k)}(x)\quad (n=0,1,\ldots)\,, $$ where $r_{n,j}$ are complex number independent of $x$. It is shown that under natural constraints, $(P_n)_{n\geq0}$ and $(Q_n)_{n\geq0}$ are semiclassical orthogonal polynomial sequences. Moreover, their corresponding moment linear functionals are related by a rational modification in the distributional sense. This leads to the concept of $\pi_N-$coherent pair with index $M$ and order $(m,k)$.