Evaluations of Derivatives of Jacobi Theta Functions in the origin

Abstract

In this article using Ramanujan's theory of Eisenstein series we evaluate completely the derivatives of the theta functions $\vartheta_1^{(2\nu+1)}(z)$ and $\vartheta_4^{(2\nu)}(z)$ in the origin in closed polynomials forms using only the first three Eisenstein series of weights 2,4, and 6.