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 $[\mathcal{Z},\mathcal{X}]=\mathcal{Z}^2+\Delta$. 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.