Nevanlinna theory of the Askey-Wilson divided difference operator

TL;DR

This paper develops a Nevanlinna theory framework based on the Askey-Wilson divided difference operator for meromorphic functions, leading to new deficiency and Picard theorems, and applications to difference equations.

Contribution

It introduces a novel Nevanlinna theory using the Askey-Wilson operator, including deficiency, Picard, and unicity theorems, and applies these to hypergeometric series and difference equations.

Findings

Defined Askey-Wilson type Nevanlinna deficiency.

Established an Askey-Wilson Picard theorem.

Applied theory to difference equations.

Abstract

This paper establishes a version of Nevanlinna theory based on Askey-Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane $\mathbb{C}$. A second main theorem that we have derived allows us to define an Askey-Wilson type Nevanlinna deficiency which gives a new interpretation that one should regard many important infinite products arising from the study of basic hypergeometric series as zero/pole-scarce. That is, their zeros/poles are indeed deficient in the sense of difference Nevanlinna theory. A natural consequence is a version of Askey-Wilosn type Picard theorem. We also give an alternative and self-contained characterisation of the kernel functions of the Askey-Wilson operator. In addition we have established a version of unicity theorem in the sense of Askey-Wilson. This paper concludes with an application to difference equations generalising the Askey-Wilson second-order divided difference equation.