 Weighted Gaussian theta sum of a complex lattice

ID: weighted-gaussian-theta-sum-of-a-complex-lattice

For an even nonnegative integer

k

θ_{k} (Λ) = \sum_{λ \in Λ} λ^{k} e^{- π ∣ λ ∣^{2}} .

Poisson summation and the Fourier eigenfunction identity for

(x + i y)^{k} e^{- π (x^{2} + y^{2})}

give

θ_{k} (Λ) = (- i)^{k} m (Λ)^{- 1} θ_{k} (Λ^{\lor}) .

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