Lattice sums for polyanalytic functions

TL;DR

This paper extends the concept of lattice sums to polyanalytic functions, providing new exact formulas and polynomial representations that relate to classical sums and have implications for isotropic composite materials.

Contribution

It introduces a novel extension of lattice sums to polyanalytic functions, deriving exact relations and computational formulas linking them to classical sums.

Findings

Established relations between polyanalytic and classical lattice sums.

Derived polynomial representations of lattice sums.

Provided new formulas involving π for isotropic composites.

Abstract

In 1892, Lord Rayleigh estimated the effective conductivity of rectangular arrays of disks and proved, by means of the Eisenstein summation, that the lattice sum $S_2$ is equal to $\pi$ for the square array. Further, it became clear that such an equality can be treated as a necessary condition of the macroscopic isotropy of composites governed by the Laplace equation. This yielded the description of two-dimensional conducting composites by the classic elliptic functions including the conditionally convergent Eisenstein series. This paper is devoted to extension of the lattice sums to double periodic polyanalytic functions. The exact relations and computationally effective formulae between the polyanalytic and classic lattice sums are established. Polynomial representations of the lattice sums are obtained. They are a source of new exact formulae for the lattice sums where the number $\pi$ persists for macroscopically isotropic composites by our suggestion.