We develop a function theory that the Wilson divided-difference 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 which miss certain hyperplanes in the sense of the Wilson operator, and formulate a Picard-type result about these holomorphic curves.