2015/02/08 by Yik‐Man Chiang, Yik-Man Chiang, Chiang, Yik-Man +2
Mathematics · #30D35 #33D99 #39A13 #39A70 #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #math.CV #msc:30D35 #msc:33D99 #msc:39A13 #msc:39A70
paper · pdf · doi:10.48550/arxiv.1502.02238
Finalised version. To appear in Advances in Mathematics
openalex publication_date 2015/02/08 · arxiv created 2018/02/03 · arxiv updated 2018/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.