Bi-orthogonal polynomial sequences and the asymmetric simple exclusion process

Abstract

We state the diffusion algebra equations of the stationary state of the three parameter (α, β and q) asymmetric simple exclusion Process as a linear functional acting on a tensor algebra. From we construct a pair of sequences, and , of monic polynomials which are bi-orthogonal, that is, they satisfy (where is a scalar). The uniqueness and existence of the pair of sequences arises from the determinant of the bi-moment matrix whose elements satisfy a pair of q-recurrence relations. The determinant is evaluated using an LDU-decomposition. If the linear functional is represented as an inner product, then the action of the polynomials Qn on the boundary vector generate a basis whose orthogonal dual vectors are given by the action of Pn on the dual boundary vector , that is . This basis gives the representation of the algebra which is associated with the Al-Salam–Chihara polynomials obtained by Sasamoto.