Wilson均差算子的Nevanlinna理论
Nevanlinna Theory of the Wilson Divided-difference Operator
摘要：
Wilson多项式位于Askey方案的顶层，被认为是最一般的超几何正交多项式。它们不是满足一个微分方程，而是满足一个由Wilson均差算子表示的二阶Sturm-Liouville型差分方程.这表明，为了更好地理解威尔逊多项式和相关主题的独特性质，应该使用相对于威尔逊算子更自然的函数理论。受Halburd和Korhonen最近工作的启发，我们建立了有穷级亚纯函数的Wilson算子的成熟的Nevanlinna理论.特别地，我们证明了关于对数导数引理的Wilson类比，这有助于我们导出Nevanlinna第二基本定理的Wilson算子版本，一些亏损关系和Picard定理.这些结果使我们对Wilson算子相关函数的零点和极点的分布有了新的认识，这与经典的观点不同。作为理论的应用，我们还得到了一个相应的五值定理和Clunie型定理，以及对数Wilson差分的逐点估计，从而得到了一些Wilson差分方程和Wilson插值方程亚纯解增长性的新估计.
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
