Bi-orthogonal polynomial sequences and the asymmetric simple exclusion process. (20th July 2015) - Brak, R; Moore, W
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 Q n on the boundary vector generate a basis whose orthogonal dual vectors are given by the action of P n 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.