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arXiv · 1511.03679

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

V. V. Borzov·E. V. Damaskinsky
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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 $μ$ and $ν$ 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 $k\geq1$.

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V. V. Borzov, E. V. Damaskinsky. 2015-11-11. Invariance of the generalized oscillator under linear transformation of the related system of orthogonal polynomials. https://arxiv.org/abs/1511.03679

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