Hypertranscendency of Perturbations of Hypertranscendental Functions

TL;DR

This paper investigates conditions under which certain meromorphic functions involving exponential and algebraic operations are hypertranscendental, providing partial solutions to Bank's conjecture and new criteria for hypertranscendence.

Contribution

It offers new partial results towards Bank's 1977 conjecture and establishes sufficient conditions for hypertranscendence of various meromorphic function combinations.

Findings

Partial solutions to Bank's conjecture on hypertranscendence of  e^h

Sufficient conditions for hypertranscendence of f+g, fg, and fg

New criteria for hypertranscendence of meromorphic functions

Abstract

Inspired by the work of Bank on the hypertranscendence of $\Gamma e^h$ where $\Gamma$ is the Euler gamma function and $h$ is an entire function, we investigate when a meromorphic function $fe^g$ cannot satisfy any algebraic differential equation over certain field of meromorphic functions, where $f$ and $g$ are meromorphic and entire on the complex plane, respectively. Our results (Theorem 1 and 2) give partial solutions to Bank's Conjecture (1977) on the hypertranscendence of $\Gamma e^h$. We also give some sufficient conditions for hypertranscendence of meromorphic function of the form $f+g$, $f\cdot g$ and $f\circ g$ in Theorem 3 and 4.