Orthogonal polynomials and the deformed Jordan plane

Abstract

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

9 pages