Denominator identity for the affine Lie superalgebra $\widehat{\mathfrak{s}\mathfrak{p}\mathfrak{o}}(2m,2m+1)$ and indefinite theta functions

In 1994, Kac and Wakimoto found the denominator identity for classical affine Lie superalgebras, generalizing that for affine Lie algebras. As an application, they obtained power series identities for some powers of $\mathrm{△}(q)$, where $\mathrm{△}(q)$ is the generating function of triangular numbers. In this article, we give a different proof of one of their identities. The main step is to prove that a certain indefinite theta function involving spherical polynomials is a modular form. We use the technique recently developed by Roehrig and Zwegers.