Abstract
Given a connected (multi)graph G, consisting of V vertices and I lines, we consider a class of multidimensional sums of the general form S G := ∑ ∞ n1=-∞ ∑ ∞ n2=-∞ ... ∑ ∞ ni=-∞ δ G (n 1 ,n 2 ,...,n I ;{N v }) / (n 2 1 + q 2 1 )(n 2 2+ q 2 2 ) ... (n 2 1 + q 2 1 ) , where the variables q i (i = 1, ... ,I) are real and positive and the variables N v (v = 1,..., V) are integer-valued. δ G (n 1 , n 2 , ..... n I ; {N v }) is a function valued in {0,1} which imposes a series of linear constraints among the summation variables n i , determined by the topology of the graph G. We prove that these sums, which we call Matsubara sums, can be explicitly evaluated by applying a G-dependent linear operator O' G to the evaluation of the integral obtained from S G by replacing the discrete variables n i by continuous real variables x i and replacing the sums by integrals.
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