Tau-functions for the Ablowitz-Ladik hierarchy: the matrix-resolvent method

We extend the matrix-resolvent method for computing logarithmic derivatives of tau-functions to the Ablowitz-Ladik hierarchy. In particular, we derive a formula for the generating series of the logarithmic derivatives of an arbitrary tau-function in terms of matrix resolvents. As an application, we provide a way of computing certain integrals over the unitary group.

Automated summary

In this study, we extended the matrix-resolvent method to compute logarithmic derivatives of tau-functions in the Ablowitz-Ladik hierarchy. We derived a formula for the generating series of logarithmic derivatives using matrix resolvents. As an application, we introduced a method to compute certain integrals over the unitary group.