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 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.Comment: Finalised version. To appear in Advances in Mathematic
