Multiple skew orthogonal polynomials and 2-component Pfaff lattice hierarchy

TL;DR

This paper introduces multiple skew-orthogonal polynomials, links them to integrable systems via Pfaffian techniques, and derives a two-component Pfaff lattice hierarchy related to the Pfaff-Toda hierarchy.

Contribution

It presents a novel class of multiple skew-orthogonal polynomials and connects them to integrable hierarchies through Pfaffian tau-functions.

Findings

Multiple skew-orthogonal polynomials can be expressed by multi-component Pfaffian tau-functions.

A two-component Pfaff lattice hierarchy is derived, related to the Pfaff-Toda hierarchy.

The work establishes connections between polynomial recurrence relations and integrable systems.

Abstract

In this paper, we introduce multiple skew-orthogonal polynomials and investigate their connections with classical integrable systems. By using Pfaffian techniques, we show that multiple skew-orthogonal polynomials can be expressed by multi-component Pfaffian tau-functions upon appropriate deformations. Moreover, a two-component Pfaff lattice hierarchy, which is equivalent to the Pfaff-Toda hierarchy studied by Takasaki, is obtained by considering the recurrence relations and Cauchy transforms of multiple skew-orthogonal polynomials.