Consider the function $f:\mathbb{T}^m\to\mathbb{R}$

$$f(\boldsymbol{x}) = \sum\limits_{{\pmb{\eta}\in\mathbb{Z}^m}}\frac{1}{1+\lambda\|\pmb{\eta}\|_{2k}^{2k} } \cos({2\pi \pmb{\eta}\cdot\pmb{x}})$$

Where $$\|\pmb{\eta}\|_{2k}^{2k} = \sum\limits_{i=1}^{m}\eta_i^{2k}$$

$k\in \mathbb{N}$ and $\lambda \in \mathbb{R}^+$

Question: Is there any closed-form expression for this summation $f(\pmb{x})$?

  • $\begingroup$ Simple answer is no. You can consult the book "Lattice sums then and now" for sums of this type. $\endgroup$ – Nemo Dec 23 '19 at 10:59

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