Two-parameter generalisations of Cauchy bi-orthogonal polynomials and integrable lattices

In this article, we consider the generalised two-parameter Cauchy two-matrix model and corresponding integrable lattice equation. It is shown that with parameters chosen as $1/k_i$ when $k_i\in\mathbb{Z}_{>0}$ ($i=1,\,2$), the average characteristic polynomials admit $(k_1+k_2+2)$-term recurrence relations, which provide us 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 example with exact solvability is demonstrated.