We are concerned with a certain class of Schlömilch series that arise naturally in the study
of diffraction problems when the scatterer is a periodic structure. By combining new results
derived from integral representations and the Poisson summation formula with known identities,
we obtain expressions which enable the series to be computed accurately and efficiently. Most
of the technical details of the derivations are omitted; they can, however, be obtained from the
technical report  available online.