# Vandermonde-type identity for Jacobi theta functions?

My question concerns an application in physics. By Vandermonde identity I refer to the following statement: take $$f_j (z)=z^j$$, where $$z=x+iy$$ is a complex coordinate and $$j$$ an integer. Make an $$N\times N$$ determinant where each column $$j$$ contains $$f_j (z_i),j=0,\ldots,N-1,i=1,\ldots,N$$. This determinant can be simply evaluated: $$\prod_{i\lt j}(z_i-z_j)$$, which is the Vandermonde formula.

Now, consider the following generalization: let the function $$f$$ be

$$f_j(x, y) = \sum_{k} e^{i B y - 1/2 (B + x)^2}$$

where the sum over $$k$$ is over all integers, $$B=2\pi j/b + k a$$, so that $$f_j$$ is explicitly periodic under $$y\mapsto y+b$$ and it is periodic up to a phase $$e^{i k a}$$ under $$x\mapsto x+a$$. All in all, the function $$f$$ is quasiperiodic over the rectangle $$a \times b$$ whose area $$ab$$ must equal $$2\pi M$$, $$M$$ being an integer. This means that $$f_{j+M}=f_{j}$$, so for fixed $$x,y$$ there are only $$M$$ distinct $$f_j(x,y)$$. This function $$f_j$$ can be related to one of the Jacobi theta functions in terms of $$z=x+iy$$.

Take $$f_j(x_i,y_i)$$ where $$j=0,\ldots,N-1; i=1,...,N$$ and arrange it into a determinant $$N\times N$$. Is there a simple formula for the value of this determinant that resembles the Vandermonde expression given above?