Holomorphic solutions of functional differential systems near singular points

Abstract

Functional analysis techniques are used to prove a theorem, analogous to the Harris-Sibuya-Weinberg theorem for ordinary differential equations, which yields as corollaries a number of existence theorems for holomorphic solutions of linear functional differential systems of the form z D y ′ ( z ) = A ( z ) y ( z ) + B ( z ) y ( α z ) + C ( z ) y ′ ( α z ) {z^D}y’(z) = A(z)y(z) + B(z)y(\alpha z) + C(z)y’(\alpha z) in the neighborhood of the singularity at z = 0 z = 0 .