Binomial series and complex difference equations

TL;DR

This paper explores the properties and convergence of binomial series in the complex domain, analyzes the growth of solutions to related difference equations, and constructs a specific Riccati difference equation with a transcendental meromorphic solution.

Contribution

It introduces new insights into binomial series convergence, growth of solutions to difference equations, and constructs a novel Riccati difference equation with a transcendental solution.

Findings

Convergence criteria for binomial series in the complex plane

Growth order of solutions to certain difference equations

Existence of a transcendental meromorphic solution of order 1/2

Abstract

We consider properties of binomial series $\sum_{n=0}^\infty a_n z^{\underline{n}}$, where $z^{\underline{n}}=z(z-1)\cdots(z-n+1)$ and the convergence of binomial series in the complex domain. The order of growth of entire and meromorphic solutions of some difference equations represented by binomial series are discussed. Examples are given. As an application, we construct a difference Riccati equation possessing a transcendental meromorphic solution of order $1/2$.