Mathematics of Computation

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 \begin{equation*} S_G:= \sum \limits _{n_1 = - \infty }^\infty \sum \limits _{n_2 = - \infty }^\infty \cdots \sum \limits _{n_I = - \infty }^\infty {\frac {{\delta _G (n_1 ,n_2 , \ldots ,n_I;\{N_v\} )}} {{\left ( {n_1^2 + q_1^2 } \right ) \left ( {n_2^2 + q_2^2 } \right ) \cdots \left ( {n_I^2 + q_I^2 } \right )}}}, \end{equation*} 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.