Yik-Man Chiang

On the meromorphic solutions of an equation of Hayman

Abstract The behavior of meromorphic solutions of differential equations has been the subject of much study. Research has concentrated on the value distribution of meromorphic solutions and their rates of growth. The purpose of the present paper is to show that a thorough search will yield a list of all meromorphic solutions of a multi-parameter ordinary differential equation introduced by Hayman. This equation does not appear to be integrable for generic choices of the parameters so we do not find all solutions—only those that are meromorphic. This is achieved by combining Wiman–Valiron theory and local series analysis. Hayman conjectured that all entire solutions of this equation are of finite order. All meromorphic solutions of this equation are shown to be either polynomials or entire functions of order one.

Difference independence of the Riemann zeta function

It is proved that the Riemann zeta function does not satisfy any nontrivial algebraic difference equation whose coefficients are meromorphic functions

An oscillation result of a third order linear differential equation with entire periodic coefficients

\phi

Subnormal solutions of non-homogeneous periodic ODEs, special functions and related polynomials

with Nevanlinna characteristic satisfying

On The Zero-Free Solutions of Linear Periodic Differential Equations In The Complex Plane

T(r, \phi)=o(r)

Oscillation results on y ″ + Ay = 0 in the complex domain with transcendental entire coefficients which have extremal deficiencies

as

Properties of analytic functions with small schwarzian derivative

r\to \infty

On the Growth of Logarithmic Difference of Meromorphic Functions and a Wiman–Valiron Estimate

We prove that the periodic equation admits a solution with finite exponent of convergence if and only if where n is a non-negative integer satisfying a certain (n + 1) × (n + 1)-determinant condition. Moreover, we obtain explicit representations for such solutions. Our result is somewhat similar to a result due to Bank, Laine and Langley [5] for a second order equation.

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

Abstract This paper offers a new and complete description of subnormal solutions of certain non-homogeneous second order periodic linear differential equations first studied by Gundersen and Steinbart in 1994. We have established a previously unknown relation that the general solutions (i.e., whether subnormal or not) of the DEs can be solved explicitly in terms of classical special functions, namely the Bessel, Lommel and Struve functions, which are important because of their numerous physical applications. In particular, we show that the subnormal solutions are written explicitly in terms of the degenerate Lommel functions Sμ, ν (ζ) and several classical special polynomials related to the Bessel functions. In fact, we solve an equivalent problem in special functions that each branch of the Lommel function Sμ, ν (ζ) degenerates if and only if Sμ, ν (e z ) has finite order of growth in ℂ. We achieve this goal by proving new properties and identities for these functions. A number of semi-classical quantization-type results are obtained as consequences. Thus our results not only recover and extend the result of Gundersen and Steinbart [Results Math. 25: 270–289, 1994], but the new identities and properties found for the Lommel functions are of independent interest in a wider context.