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

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 $$Q_n(x)=P_n(x)+a_1{P}_{n-1}(x)+...+a_k{P}_{n-k}(x).$$ 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 $k\ge 1$.

10 pages, 0 figures, The work is based on the report presented at the international conference MQFT 2015

References in corpus (1)

• On the dimensions of the oscillator algebras induced by orthogonal polynomials