Orthogonal polynomials and the deformed Jordan plane

TL;DR

This paper explores a unital algebra with two generators, revealing connections between its representations and classical orthogonal polynomials like Jacobi and Hahn, depending on a parameter.

Contribution

It constructs irreducible tridiagonal representations of the algebra, linking them to well-known orthogonal polynomials based on the parameter value.

Findings

Representations linked to para-Krawtchouk, Hahn, and Jacobi polynomials

Parameter-dependent classification of algebra representations

Explicit construction of irreducible tridiagonal matrices

Abstract

We consider the unital associative algebra $\mathcal{A}$ with two generators $\mathcal{X}$, $\mathcal{Z}$ obeying the defining relation $[\mathcal{Z},\mathcal{X}]=\mathcal{Z}^2+\Delta$. We construct irreducible tridiagonal representations of $\mathcal{A}$. Depending on the value of the parameter $\Delta$, these representations are associated to the Jacobi matrices of the para-Krawtchouk, continuous Hahn, Hahn or Jacobi polynomials.