Closed-Form Nonrecurrent Formulas for the Coefficients of the Taylor Series of the Weierstrass Sigma Function

We obtain two closed-form nonrecurrent formulas for the coefficients of the Taylor series of the Weierstrass sigma function related to elliptic function theory. These results are derived from the differential-recurrence and two-dimensional recurrence relations that arise when searching for the analytic solutions of the system of partial differential equations constructed by Weierstrass for the sigma function.