On the Expansion Coefficients of Tau-function of the BKP Hierarchy

TL;DR

This paper investigates the expansion coefficients of the BKP hierarchy tau function, extending known formulas to cases where the tau function vanishes at zero, thereby broadening the understanding of BKP solutions.

Contribution

It generalizes the Giambelli-type formula for BKP tau functions to include the case when the tau function equals zero at the origin.

Findings

Derived a generalized formula for BKP tau function coefficients at zero

Extended the applicability of Schur Q-function expansions

Provided new insights into BKP hierarchy solutions

Abstract

We study the series expansion of the tau function of the BKP hierarchy applying the addition formulae of the BKP hierarchy. Any formal power series can be expanded in terms of Schur functions. It is known that, under the condition $\tau(x)\neq0$, a formal power series $\tau(x)$ is a solution of the KP hierarchy if and only if its coefficients of Schur function expansion are given by the so called Giambelli type formula. A similar result is known for the BKP hierarchy with respect to Schur's Q-function expansion under a similar condition. In this paper we generalize this result to the case of $\tau(0)=0$.