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.

中文翻译：

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Cauchy双正交多项式和可积格的两参数推广

在本文中，我们考虑了广义两参数柯西两矩阵模型和相应的可积晶格方程。结果表明，选择参数为\（1 / k_i \），\（k_i \在{\ mathbb {Z}} _ {> 0} \）（\（i = 1，\，2 \））中，平均特征多项式允许\（（k_1 + k_2 + 2）\）-项递推关系，可以将其解释为可积晶格的谱问题。然后，tau函数由广义Cauchy两矩阵模型的分区函数以及Gram行列式给出。给出了最简单的可解决的例子。