Nevanlinna Theory of the Wilson Divided-difference Operator

TL;DR

This paper develops a Nevanlinna theory tailored for the Wilson divided-difference operator, providing new tools to analyze the value distribution and growth of meromorphic functions related to Wilson polynomials.

Contribution

It introduces a comprehensive Nevanlinna theory for the Wilson operator, including analogues of classical theorems and new growth estimates for related difference equations.

Findings

Established Wilson analogue of the lemma on logarithmic derivatives

Derived Wilson versions of Nevanlinna's Second Fundamental Theorem and Picard's Theorem

Provided growth estimates for meromorphic solutions of Wilson difference equations

Abstract

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.