Properties of meromorphic solutions of certain equations via Nevanlinna theory

Abstract

Special functions are functions encountered in the process of solving various problems in physics. In a natural way one of the directions concerning analysis of properties of special function is to examine them as solutions of linear differential equations. This line of thought can be extended not only to nonlinear ordinary differential equations, but also to difference equations and delay-differential equations. Well known examples of nonlinear ordinary differential equations with solutions expressed in terms of special functions include Riccati equations (Airy functions) and Painlevé equations (Airy functions for P 2 P_2 , Bessel functions for P 3 P_3 or Whittaker functions for P 4 P_4 ). Painlevé transcendents have been thoroughly examined by means of global Nevanlinna theory, resulting in estimates of growth, descriptions of distribution of zeros and poles, analysis of existence of exceptional and ramified values. It has also provided insight into behaviour of special functions involved. Recent development of difference Nevanlinna theory opened similar possibilities of analysis of behaviour of solutions of difference and delay-differential equations in general, and various involved special functions in particular.