Derivative formulas for Hermitian theta functions of degree two (Q2748314)

Let \(Z\) be a symmetric complex matrix with positive definite imaginary part, \(p\) a point of the Siegel upper half-space \(S_g\) of degree \(g\), \(x\in\mathbb C^g\) a complex vector and \(m\) a characteristic, then the seriesNEWLINENEWLINE\[NEWLINE\Theta[m](Z| x)=\sum_{p\in\mathbb Z^g}e\bigl(\frac 12(p+m')^tZ(p+m')+(p+m')^t(x+m'')\bigr)NEWLINE\]NEWLINE is the Riemann's theta-function of characteristic \(m\) with module \(Z\). In case of \(g=1\), there is a Jacobi derivative formula for this theta-function. In this paper the author generalizes it to the case \(g=2\).