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Alternative summation orders for the Eisenstein series \(G_2\) and Weierstrass \(\wp\)-function. (English) Zbl 1455.30004

The authors study the Eisenstein \(G_2\) series \[G_2(\tau)=\sum_{m}\left[\sum_{n}\,\frac{1}{(n+m\tau)^2}\right],\] with \(m,n\in\mathbb{Z},\,(m,n)\not= (0,0)\), and the Weierstrass \(\wp\)-function \[\wp(z)=\frac{1}{z^2}+\sum_{(m,n)\in\mathbb{Z}^2\setminus \{(0,0)\}}\,\left[\frac{1}{(z+n+m\tau)^2}-\frac{1}{(n+m\tau)^2}\right]\] \(\tau\in\mathbb{H},z\not\in \mathbb{Z}\tau+\mathbb{Z}\), in the context of conditionally convergent series and their behavior under change in the order of summation.
To this end they introduce a set of shapes as follows.
Let \(\mathcal{K}\) be the class of compact sets \(K\in\mathbb{R}^2\) that are convex, have non-empty interior, and are symmetric about the \(x\)- and \(y\)-axes. Define:
1. For each \(K\in\mathcal{K}\), let \(h_k\) be the real-valued function whose graph is the upper boundary of the shape \(K\). The functions \(h_k\) is compactly supported on an interval \([-A,A]\), is an even function, and its reflection \(-h_k\) is the lower boundary of \(K\).
2. For a shape \(K\in\mathcal{K}\) and an array \((a_{m,n})_{m,n\in\mathbb{Z}}\) of complex numbers, define \[\sum_K\,a_{m,n}=\lim_{\lambda\rightarrow\infty}\,\sum_{(m,n)\in(\lambda K)\cap\mathbf{Z}^2}\,a_{m,n},\] provided the limit exists. The authors refer to this sum as the \(K\)-summation, or the shape summation with respect to the shape \(K\), of the array \((a_{m,n})\).
3.If \(K\in\mathcal{K}\), denote by \(G_2(K,\tau)\) the \(K\)-summation of the weight-2 Eisenstein series, defined by \[G_2(K,\tau)=\sum_{K}\,\frac{1}{(m\tau +n)^2},\] provided the limit defining the summation exists, and with the convention that \(a_{0,0}=0\).
4. If \(K\in\mathcal{K}\) and \(G_2(K<\tau)\) is defined, the authors denote by \(E(K,\tau)\) the residual function associated to \(K\), defined as \[E(K,\tau)=G_2(K,\tau)-G_2(\tau).\]
The main results are the following.
Theorem 5. For all \(\tau\in\mathbb{H}\) and all \(K\in\mathcal{K}\), the limit defining \(G_2(K,\tau)\) exists. The residual function is given by \[E(K,\tau)=4\int_0^A\,\frac{h_K(x)}{\tau^2x^2-h_K^2(x)}dx,\] where, as before, \(A\) denotes a number for which \(h_K\) is supported on \([-A,A]\).
Proposition 9. For \(K\in\mathcal{K},\tau\in\mathbb{H}\) and \(z\not\in \mathbb{Z}\tau +\mathbb{Z}\), the \(K\)-summation \(\wp(K,\tau)\) is defined and satisfies \[\wp(K,z)=\wp(z)+G_2(\tau)+E(K,\tau).\]

MSC:

30B99 Series expansions of functions of one complex variable
40A05 Convergence and divergence of series and sequences
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