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

Abstract

We consider the families of polynomials P = { P_n(x)}_n=0^∞ and Q = { Q_n(x)}_n=0^∞ orthogonal on the real line with respect to the respective probability measures μ and ν. We assume that { Q_n(x)}_n=0^∞ and {P_n(x)}_n=0^∞ are connected by linear relations. In the case k = 2, we describe all pairs (P,Q) for which the algebras A_P and A_Q of generalized oscillators generated by { Qn(x)}_n=0^∞ and { Pn(x)}_n=0^∞ coincide. We construct generalized oscillators corresponding to pairs (P,Q) for arbitrary k ≥ 1.