On a generalization of the Fay-Sato identity for KP Baker functions and its application to constrained hierarchies

TL;DR

This paper extends the Fay-Sato identity for KP Baker functions, deriving new formulas from the differential Fay identity, and applies them to constrained hierarchies, providing determinant identities and a universal differential equation.

Contribution

It introduces a generalized Fay-Sato identity for KP Baker functions and develops new formulas and recurrence relations for constrained hierarchies.

Findings

Derived new formulas for KP hierarchy from the differential Fay identity

Established determinant identities for k-constrained hierarchies

Presented explicit formulas for k=1,2,3 and recurrence relations for others

Abstract

Some new formulas for the KP hierarchy are derived from the differential Fay identity. They proved to be useful for the $k$-constrained hierarchies providing a series of determinant identities for them. A differential equation is introduced which is called ``universal" since it plays an important role for all the $k$-constrained hierarchies. In the cases $k=1,2$ and 3 explicit formulas are presented, in all the others recurrence relations are given which enable one to obtain the identities.