Differential equations involving circulant matrices

Introduction.This paper develops a theory for the solution of ordinary and partial differential equations whose structure involves the algebra of circulants.Recent interest of circulants is evident in a book by Davis [1].This paper shows how the algebra of 2 x 2 circulants relates to the study of the harmonic oscillator, the Cauchy-Riemann equations, Laplace's equation, the Lorentz transformation, and the wave equation.It then uses n x n circulants to suggest natural generalizations of these equations to higher dimensions. The algebra of circulants.An n x n circulant is a matrix of the form X » Note that X has arbitrary entries x 0 , x h ..., x n _x in the top row and the entries are moved over one place to the right in each succeeding row.Let K denote the circulant with x x = 1 and jc y = 0 for all j' ^ 1.Then the arbitrary circulant Xequals £g=J x h K h , and K n = /.[K° = /also.]Define complex circulants E Q , E Ì9 ..., E n _ x by