Two-Parameter Generalizations of Cauchy Bi-Orthogonal Polynomials and Integrable Lattices

Abstract

In this article, we consider the generalised two-parameter Cauchy two-matrix model and the corresponding integrable lattice equation. It is shown that with parameters chosen as $$1/k_i$$ , $$k_i\in {\mathbb {Z}}_{>0}$$ ( $$i=1,\,2$$ ), the average characteristic polynomials admit $$(k_1+k_2+2)$$ -term recurrence relations, which can be interpreted as spectral problems for integrable lattices. The tau function is then given by the partition function of the generalised Cauchy two-matrix model as well as Gram determinant. The simplest solvable example is given.