Bi-orthogonal Polynomials and the Five parameter Asymmetric Simple Exclusion Process

We apply the bi-moment determinant method to compute a representation of the matrix product algebra -- a quadratic algebra satisfied by the operators $\mathbf{d}$ and $\mathbf{e}$ -- for the five parameter ($\alpha$, $\beta$, $\gamma$, $\delta$ and $q$) Asymmetric Simple Exclusion Process. This method requires an $LDU$ decomposition of the ``bi-moment matrix''. The decomposition defines a new pair of basis vectors sets, the `boundary basis'. This basis is defined by the action of polynomials $\{P_n\}$ and $\{Q_n\}$ on the quantum oscillator basis (and its dual). Theses polynomials are orthogonal to themselves (ie.\ each satisfy a three term recurrence relation) and are orthogonal to each other (with respect to the same linear functional defining the stationary state). Hence termed `bi-orthogonal'. With respect to the boundary basis the bi-moment matrix is diagonal and the representation of the operator $\mathbf{d}+\mathbf{e}$ is tri-diagonal. This tri-diagonal matrix defines another set of orthogonal polynomials very closely related to the the Askey-Wilson polynomials (they have the same moments).