# $C_\lambda$- Extended oscillator algebra and $d$-orthogonal polynomials

## TL;DR

This paper constructs an analytic realization of the $C_$-extended oscillator algebra using difference-differential operators and introduces new $d$-orthogonal polynomials extending Hermite and Laguerre polynomials, analyzing their properties and matrix realizations.

## Contribution

It provides a systematic construction of $d$-orthogonal polynomials related to the $C_$-extended oscillator algebra and explores their algebraic and analytical properties.

## Key findings

- Derived recurrence relations and difference-differential equations for the polynomials.
- Established raising and lowering operators for the polynomial families.
- Constructed matrix realizations of the algebra using these 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.