  We reformulate the Corteel-Williams equations for the stationary state of the
two parameter Asymmetric Simple Exclusion Process (TASEP) as a linear map
$\mathcal{L}(\,\cdot\,)$, acting on a tensor algebra built from a rank two free
module with basis $\{e_1,e_2\}$. From this formulation we construct a pair of
sequences, $\{P_n(e_1)\}$ and $\{Q_m(e_2)\}$, of bi-orthogonal polynomials
(BiOPS), that is, they satisfy $\mathcal{L}(P_n(e_1)\otimes
Q_m(e_2))=\Lambda_n\delta_{n,m}$. The existence of the sequences arises from
the determinant of a Pascal triangle like matrix of polynomials. The
polynomials satisfy first order (uncoupled) recurrence relations. We show that
the two first moments $\mathcal{L}(P_n\, e_1\, Q_m)$ and $\mathcal{L}(P_n\,
e_2\, Q_m)$ give rise to a matrix representation of the ASEP diffusion algebra
and hence provide an understanding of the origin of the matrix product Ansatz.
The second moment $\mathcal{L}(P_n\, e_1 e_2\,Q_m )$ defines a tridiagonal
matrix which makes the connection with Chebyshev-like orthogonal polynomials.
