Alternative summation orders for the Eisenstein series $G_2$ and Weierstrass $\wp$-function

Abstract

We consider alternative orders of summation for the conditionally convergent series defining the weight-2 Eisenstein series $G_2$ and the Weierstrass $\wp$-function. The resulting sums differ from the standard ones by a residual term that can be thought of as a function of the shapes with respect to which we sum. We compute this residual function explicitly and give some examples. The results generalize the well-known quasimodularity relationship between $G_2$ and its series summed in the reverse order.