Evaluation and regularization of phase-modulated Eisenstein series and application to double Schlomilch-type sums


In the study of periodic media, conditionally convergent series are frequently encountered and their regularization is crucial for applications. We derive an identity that regularizes two-dimensional generalized Eisenstein series for all Bravais lattices, yielding physically meaningful values. We also obtain explicit forms for the generalized series in terms of conventional Eisenstein series, enabling their closed-form evaluation for important high symmetry lattices. Results are then used to obtain representations for the related cylindrical harmonic sums, which are also given for all Bravais lattices. Finally, we treat displaced lattices of high symmetry, expressing them in terms of origin-centered lattices via geometric multi-set identities. These identities apply to all classes of two-dimensional sums, allowing sums to be evaluated over each constituent of a unit cell that possesses multiple inclusions.

Journal of Mathematical Physics, 59:072902