On Matrix Product Ansatz for Asymmetric Simple Exclusion Process with Open Boundary in the Singular Case

We study a substitute for the matrix product ansatz for asymmetric simple exclusion process with open boundary in the "singular case" alpha beta = q(N)gamma(delta), when the standard form of the matrix product ansatz of Derrida et al. (J Phys A 26(7):1493-1517, 1993) does not apply. In our approach, the matrix product ansatz is replaced with a pair of linear functionals on an abstract algebra. One of the functionals, phi(1), is defined on the entire algebra, and determines stationary probabilities for large systems on L >= N + 1 sites. The other functional, phi(0), is defined only on a finite-dimensional linear subspace of the algebra, and determines stationary probabilities for small systems on L < N + 1 sites. Functional phi(0) vanishes on non-constant Askey-Wilson polynomials and in non-singular case becomes an orthogonality functional for the Askey-Wilson polynomials.