Finite Rank Solutions of Euler-Poisson-Darboux Equations

The subject of this article are so-called finite rank solutions originally introduced by Laplace for second-order linear partial differential equations (pde's) in the plane. They consist of linear combinations of undetermined functions and their derivatives up to a certain order called their rank. In this article a method is proposed in order to determine finite rank solutions for linear pde's of any order in any number of independent variables. It is worked out in detail for Euler-Poisson-Darboux equations in one, two or three space variables, and numerous solutions are explicitly given. The extension of this proceeding to general linear pde's are discussed.