Two generalizations of Jacobi's derivative formula

In this paper we generalize Jacobi's derivative formula, considered as an identity for theta functions with characteristics and their derivatives, to higher genus/dimension. By applying the methods developed in our previous paper [GSM04], several generalizations to Siegel modular forms are obtained. These generalizations are identities satisfied by theta functions with characteristics and their derivatives at zero. Equating all the coefficients of the Fourier expansion of these relations to zero yields non-trivial combinatorial identities.