On representations of generalized oscillator for two sequences of linearly related orthogonal polynomials

We consider two families of monic polynomials P = {Pn(x)}∞n=0 and Q = {Qn(x)}∞n=0 orthogonal with respect to probability measures μ and v on the real line, respectively. Let {Qn(x)}∞n=0 and {Pn(x)}∞n=0 be connected by the relations Qn(x) = Pn(x) + a1Pn-1(x). We consider a generalized oscillator algebras Ap and Aq associated with the sequences P and Q. In the article we describe all the pairs (P, Q) for which Ap = AQ and construct the generalized oscillator algebras.