\(C_\lambda\)- Extended oscillator algebra and \(d\)-orthogonal polynomials

Abstract

In this paper we first construct an analytic realization of the \(C_\lambda\)-extended oscillator algebra with the help of difference-differential operators. Secondly, we study families of \(d\)-orthogonal polynomials which are extensions of the Hermite and Laguerre polynomials. The underlying algebraic framework allowed us a systematic derivation of their main properties such as recurrence relations, difference-differential equations, lowering and rising operators and generating functions. Finally, we use these polynomials to construct a realization of the \(C_\lambda\)-extended oscillator by block matrices.