Biorthogonal Polynomial Sequences and the Asymmetric Simple Exclusion Process

Abstract

The diffusion algebra equations of the stationary state of the three parameter Asymmetric Simple Exclusion Process are represented as a linear functional, acting on a tensor algebra. From the linear functional, a pair of sequences (P and Q) of monic polynomials are constructed which are bi-orthogonal, that is, they are orthogonal with respect to each other and not necessarily themselves. 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 action of the linear functional is represented as an inner product, then the actio..