Invariance of the generalized oscillator under linear transformation of the related system of orthogonal polynomials

TL;DR

This paper investigates the invariance of generalized oscillator algebras associated with orthogonal polynomial families under linear transformations, providing a complete characterization for the case when the transformation involves two polynomials and extending to arbitrary degrees.

Contribution

It characterizes when two families of orthogonal polynomials related by a linear transformation have identical generalized oscillator algebras, and constructs these algebras for any polynomial degree.

Findings

For k=2, all pairs of polynomial families with equal oscillator algebras are described.

Constructs generalized oscillator algebras for polynomial families related by linear transformations for any k≥1.

Provides conditions under which the algebraic structures remain invariant under polynomial transformations.

Abstract

We consider two families of polynomials $\mathbb{P}=\polP$ and $\mathbb{Q}=\polQ$\footnote{Here and below we consider only monic polynomials.} orthogonal on the real line with respect to probability measures $\mu$ and $\nu$ respectively. Let $\polQ$ and $\polP$ connected by the linear relations $$ Q_n(x)=P_n(x)+a_1P_{n-1}(x)+...+a_kP_{n-k}(x).$$ Let us denote $\mathfrak{A}_P$ and $\mathfrak{A}_Q$ generalized oscillator algebras associated with the sequences $\mathbb{P}$ and $\mathbb{Q}$. In the case $k=2$ we describe all pairs ($\mathbb{P}$,$\mathbb{Q}$), for which the algebras $\mathfrak{A}_P$ and $\mathfrak{A}_Q$ are equal. In addition, we construct corresponding algebras of generalized oscillators for arbitrary $k\geq1$.