Fundamental solution of a multidimensional difference equation with periodical and matrix coefficients (Q1345330)

The paper deals with partial difference equations of the form \[ \sum_{p\in \mathbb{Z}^d} b(p, \ell) y(\ell- p)= x(\ell), \qquad \ell\in \mathbb{Z}^d \] where \(x,y: \mathbb{Z}^d\to \mathbb{C}\) and \(b: \mathbb{Z}^d \times \mathbb{Z}^d\to \mathbb{C}\). The function \(b(p, \ell)\) is supposed to be periodic with respect to the second variable and to vanish for all but finitely many \(p\in \mathbb{Z}^d\). After a transformation of the given equation to an equation for a vector sequence some integral formulae for the fundamental solution of the latter equation are derived. Those formulae are applied in order to get some results about the asymptotic properties of the fundamental solution and to study a model equation on a hexagonal grid.