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