Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 175-209
The Hong Kong University of Science and Technology,
Department of Mathematics
Clear Water Bay, Kowloon, Hong Kong;
henry.cheng 'at' family.ust.hk
The Hong Kong University of Science and Technology,
Department of Mathematics
Clear Water Bay, Kowloon, Hong Kong;
machiang 'at' ust.hk
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.
2010 Mathematics Subject Classification: Primary 30D35; Secondary 30D30, 33C45, 39A05.
Key words: Wilson divided-difference operator, complex function theory, Poisson–Jensen formula, Nevanlinna theory.
Reference to this article: K. H. Cheng and Y.-M. Chiang: Nevanlinna theory of the Wilson divided-difference operator. Ann. Acad. Sci. Fenn. Math. 42 (2017), 175-209.
https://doi.org/10.5186/aasfm.2017.4211
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