Partial-skew-orthogonal polynomials and related integrable lattices with Pfaffian tau-functions

TL;DR

This paper introduces partial-skew-orthogonal polynomials (PSOPs), explores their connection to integrable lattices with Pfaffian tau-functions, and reveals new integrable systems related to random matrix ensembles and vector Padé approximants.

Contribution

It proposes PSOPs as a modification of SOPs, derives nine integrable lattices with Pfaffian tau-functions, and establishes novel links between integrable systems and vector Padé approximants.

Findings

Derived nine integrable lattices with Pfaffian tau-functions.

Connected integrable lattices to the Bures random matrix ensemble.

First example linking integrable lattices to vector Padé approximants.

Abstract

Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in [22], we propose the concept of partial-skew-orthogonal polynomials (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the tau-functions also admit the representations of multiple integrals. Among these integrable lattices, some of them are known, while the others are novel to the best of our knowledge. In particular, one integrable lattice is related to the partition function of the Bures random matrix ensemble. Besides, we derive a discrete integrable lattice, which can be used to compute certain vector Pad\'e approximants. This yields the first example regarding the connection between integrable lattices and vector Pad\'e approximants, for which Hietarinta, Joshi and Nijhoff pointed out that "This field remains largely to be explored." in the recent monograph [27, Section 4.4] .