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

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.