In 1992, Gundersen proposed the following famous question: suppose that two distinct and non-constant meromorphic functions f and g in the complex plane share 0, 1, c IM and ∞ CM, where c is a finite complex number such that c∉0, 1. Do f and g share 0, 1, c, ∞ CM? Based upon the assumption that f and g are two distinct and non-constant meromorphic functions of finite order and the assumption that c satisfies certain conditions, we have studied this question and proved the following result in this paper: suppose that f and g are two distinct and non-constant meromorphic functions of finite order such that they share 0, 1, c, ∞ IM, the positive integer M is defined as M=max{|C|: C∈C₀}, where C₀ is a set of all the finite critical values of f(z), that is to say C₀={C∈C: there exists a z∈C, such that f(z)=C and f′(z)=0}. Then when |c|>M, f and g share all the four values 0, 1, c, ∞ CM. As a corollary of this result, we give an affirmative answer to the above Gundersen′s question under the assumption that f and g are two distinct and non-constant meromorphic functions of finite order and the assumption that c satisfies certain conditions.

This paper studies the algebraic independence of Euler gamma function Γ and any one of a class $\tilde{S}$ of L-functions satisfying the axioms (Ⅰ)—(Ⅴ), and proves that the Euler gamma function Γ and any one of a class $\tilde{S}$ of L-functions do not satisfy any given non-trivial algebraic differential equation. The main result of this paper improves and extends the corresponding results by Li and Ye in 2016 and by Chen and Wang in 2021 respectively.

In this paper, we study a uniqueness question of transcendental entire functions sharing two distinct rational functions IM with their k-th order derivatives. The main result obtained in this paper extends the corresponding result concerning non-constant entire functions sharing two distinct small functions IM with their first order derivatives, which was proved by Qiu Gangdi in 2000, and gives an affirmative answer to a conjecture proposed by Li Ping and Yang Chungchun in 1999 based upon the assunption that transcendental entire functions sharing two distinct rational functions IM with their k-th order derivatives.