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

$$S_G:= \sum\limits_{n_1 = - \infty }^\infty \sum\limits_{n_2 = - \... ...ht) \left( {n_2^2 + q_2^2 } \right) \cdots \left( {n_I^2 + q_I^2 } \right)}}},$$

where the variables $q_i$ ( $i=1,\ldots,I$) are real and positive and the variables $N_v$ ( $v=1,\ldots,V$) are integer-valued. $\delta_G(n_1 ,n_2 , \ldots ,n_I;\{N_v\} )$ 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 $\hat{\mathcal{O}}'_G$ 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.