Degenerate Addition Formulas of the KP Hierarchy and Applications

It is well known that tau functions of the KP hierarchy satisfy addition formulas. Among them we consider the formula which expresses a tau function with shifted arguments by $2n$ parameters in terms of the same tau function with shifted arguments by two parameters in the form of determinant. We then take the limits of it tending some of parameters to zero. As an important special case we obtain the formula which connects the shifted tau function to the Wronskian of functions obtained by substituting parameter values into the spectral variable of the wave function. As an application, we prove the equivalence of vertex operators and Darboux transformations. As another application, we derive a new addition formula for Riemann's theta functions of Riemann surfaces by considering theta function solutions of the KP hierarchy. It can be considered as a limit of Fay's famous formula (43) in the book [Lecture Notes in Math., Vol. 352, Springer, Berlin, 1973]. But the formula we have derived is a different limit from the limits of (43) considered in that book.