The inverse problem for linearly related orthogonal polynomials: General case

TL;DR

This paper investigates the inverse problem in orthogonal polynomial theory, focusing on two polynomial families linked by a linear algebraic relation, extending understanding of their structural connections.

Contribution

It generalizes the inverse problem for orthogonal polynomials to cases with arbitrary linear relations between two polynomial families.

Findings

Derived conditions for the inverse problem in the general case

Extended the theory to include arbitrary linear relations

Provided a framework for analyzing related orthogonal polynomial pairs

Abstract

We study the inverse problem in the theory of (standard) orthogonal polynomials involving two polynomials families $(P_n)_n$ and $(Q_n)_n$ which are connected by a linear algebraic structure such as $$P_n(x)+\sum_{i=1}^N r_{i,n}P_{n-i}(x)=Q_n(x)+\sum_{i=1}^M s_{i,n}Q_{n-i}(x)$$ for all $n=0,1, \dots$ where $N$ and $M$ are arbitrary nonnegative integer numbers.