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Value sharing and Stirling numbers

Aimo Hinkkanen, Ilpo Laine

Abstract

Let $f$ be an entire function and $L(f)$ a linear differential polynomial in $f$ with constant coefficients. Suppose that $f$, $f'$, and $L(f)$ share a meromorphic function $α(z)$ that is a small function with respect to $f$. A characterization of the possibilities that may arise was recently obtained by Lahiri. However, one case leaves open many possibilities. We show that this case has more structure than might have been expected, and that a more detailed study of this case involves, among other things, Stirling numbers of the first and second kinds. We prove that the function $α$ must satisfy a linear homogeneous differential equation with specific coefficients involving only three free parameters, and then $f$ can be obtained from each solution. Examples suggest that only rarely do single-valued solutions $α(z)$ exist, and even then they are not always small functions for $f$.

Value sharing and Stirling numbers

Abstract

Let be an entire function and a linear differential polynomial in with constant coefficients. Suppose that , , and share a meromorphic function that is a small function with respect to . A characterization of the possibilities that may arise was recently obtained by Lahiri. However, one case leaves open many possibilities. We show that this case has more structure than might have been expected, and that a more detailed study of this case involves, among other things, Stirling numbers of the first and second kinds. We prove that the function must satisfy a linear homogeneous differential equation with specific coefficients involving only three free parameters, and then can be obtained from each solution. Examples suggest that only rarely do single-valued solutions exist, and even then they are not always small functions for .
Paper Structure (6 sections, 6 theorems, 176 equations)

This paper contains 6 sections, 6 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Initial remarks on Lahiri's work
  3. Stirling numbers
  4. The formula for the derivatives of $f$
  5. The differential equation for $\alpha(z)$
  6. Solving the differential equation for $\alpha$

Key Result

Theorem 1.1

Suppose that $f$ is a non-constant entire function, $\alpha (z)$ is a non-zero small function with respect to $f$, and $n\geq 2$ is an integer. Let $L(f):=a_{n}f^{(n)}+a_{n-1}f^{(n-1}+\cdots +a_{1}f'+a_{0}f$ be a linear differential polynomial to $f$ with constant coefficients such that $a_{0}\neq 0 where $b\neq 0$ is a constant. (iii) $f(z)=\lambda e^{cz}+p(z)+a(z),$ where $c$ is a constant, $p(z

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 5.1