We develop a function theory that the Wilson divideddifference operator gives rise to. Firstly, we study entire functions by developing a Wiman-Valiron theory for a polynomial series whose basis interacts nicely with the Wilson operator. Secondly, we extend this polynomial basis so as to obtain a Laurent-type series, and formulate a residue theory regarding the functions defined by this kind of series. Thirdly, we study meromorphic functions by giving a Nevanlinna theory of the Wilson operator, and we obtain a Wilson analogue of the little Picard theorem. Finally, we investigate the images of holomorphic curves in the complex projective space C Pn which miss certain hyperplanes in the sense of the Wilson operator, and formulate a Picard-type result about these holomorphic curves.
