  Sitting at the top level of the Askey-scheme, Wilson polynomials are regarded
as the most general hypergeometric orthogonal polynomials. Instead of a
differential equation, they satisfy a second order Sturm-Liouville type
difference equation in terms of the Wilson divided-difference operator. This
suggests that in order to better understand the distinctive properties of
Wilson polynomials and related topics, one should use a function theory that is
more natural with respect to the Wilson operator. Inspired by the recent work
of Halburd and Korhonen, we establish a full-fledged Nevanlinna theory of the
Wilson operator for meromorphic functions of finite order. In particular, we
prove a Wilson analogue of the lemma on logarithmic derivatives, which helps us
to derive Wilson operator versions of Nevanlinna's Second Fundamental Theorem,
some defect relations and Picard's Theorem. These allow us to gain new insights
on the distributions of zeros and poles of functions related to the Wilson
operator, which is different from the classical viewpoint. We have also
obtained a relevant five-value theorem and Clunie type theorem as applications
of our theory, as well as a pointwise estimate of the logarithmic Wilson
difference, which yields new estimates to the growth of meromorphic solutions
to some Wilson difference equations and Wilson interpolation equations.
