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JOURNALS // Teoreticheskaya i Matematicheskaya Fizika // Archive

TMF, 2017 Volume 190, Number 2, Pages 267–276 (Mi tmf9118)

This article is cited in 2 papers

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

V. V. Borzova, E. V. Damaskinskyb

a St. Petersburg State University of Telecommunications, St. Petersburg, Russia
b Military Institute (Engineering-Technical), Military Academy of Materiel and Technical Security, St. Petersburg, Russia

Abstract: We consider the families of polynomials $\mathbb P=\{P_n(x)\}_{n=0}^\infty$ and $\mathbb Q=\{Q_n(x)\}_{n=0}^\infty$ orthogonal on the real line with respect to the respective probability measures $\mu$ and $\nu$. We assume that $\{Q_n(x)\}_{n=0}^\infty$ and $\{P_n(x)\}_{n=0}^\infty$ are connected by linear relations. In the case $k=2$, we describe all pairs $(\mathbb P,\mathbb Q)$ for which the algebras $\mathfrak A_P$ and $\mathfrak A_Q$ of generalized oscillators generated by $\{Q_n(x)\}_{n=0}^\infty$ and $\{P_n(x)\}_{n=0}^\infty$ coincide. We construct generalized oscillators corresponding to pairs $(\mathbb P,\mathbb Q)$ for arbitrary $k\ge1$.

Keywords: generalized oscillator, orthogonal polynomial.

Received: 09.12.2015
Revised: 10.04.2016

DOI: 10.4213/tmf9118


 English version:
Theoretical and Mathematical Physics, 2017, 190:2, 228–236

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