General relations between the derivatives of the third jacobi theta function

In this article we show how symmetry allows the derivation of an in_nite number of exact relations between the derivatives of the third Jacobi theta function, which is a special kind of series. Despite the transcendental nature of the theta function in general, the relations involve linear combinations with relatively simple integer coe_cients. One corollary of the equations is an exact value for the _rst order derivative. Linear dependency of the equations prevents an explicit solution for the higher order derivatives, but we succeed in transforming the equations to a new elegant form that comes as close as it gets to a solution.