On the evaluation of Matsubara sums

Given a connected (multi)graph G, consisting of V vertices and I lines, we consider a class of multidimensional sums of the general form SG := ∞ ∑ n1=−∞ ∞ ∑ n2=−∞ · · · ∞ ∑ nI=−∞ δG(n1, n2, . . . , nI ; {Nv}) ( n1 + q 2 1 ) ( n2 + q 2 2 ) · · · ( nI + q 2 I ) , where the variables qi (i = 1, . . . , I) are real and positive and the variables Nv (v = 1, . . . , V ) are integer-valued. δG(n1, n2, . . . , nI ; {Nv}) is a function valued in {0, 1} which imposes a series of linear constraints among the summation variables ni, 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 Ô′ G to the evaluation of the integral obtained from SG by replacing the discrete variables ni by continuous real variables xi and replacing the sums by integrals.