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We present a new definition of Euler Gamma function. From the complex analysis and transalgebraic viewpoint, it is a natural characterization in the space of finite order meromorphic functions. We show how the classical theory and formulas…
We study vanishing cycles naturally attached to a meromorphic function with isolated singularities, in both local and global settings.
We investigate the existence and distribution of Herman rings of transcendental meromorphic functions which have at least one omitted value. If all the poles of such a function are multiple then it has no Herman ring. Herman rings of period…
In this paper we define a continuous version of multiple zeta functions. They can be analytically continued to meromorphic functions on $\mathbb{C}^r$ with only simple poles at some special hyperplanes. The evaluations of these functions at…
We prove that if D is a domain in C, alpha>1 and c>0, then the family F of functions meromorphic in D such that |f'(z)|/(1+|f(z)|^alpha)>c for every z in D is normalin D. For alpha=1, the same assumptions imply quasi-normality but not…
We consider the set of power functions defined on the set of positive real number, and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative…
The Mordell-Tornheim zeta function and the Herglotz-Zagier function $F(x)$ are two important functions in Mathematics. By generalizing a special case of the former, namely $\Theta(z, x)$, we show that the theories of these functions are…
In this paper, we investigate the value distribution for linear q-difference polynomials of transcendental meromorphic functions of zero order which improves the results of Xu, Liu and Cao (\cite{Xu & Liu & Cao & 2015}). We also investigate…
The subject of this paper is the bounded level curves of a meromorphic function $f$ with domain $G$ such that each component of $\partial{G}$ consists of a level curve of $f$. (A primary example of such a function being a ratio of finite…
The aim of this paper is to establish properties of the solutions to the $\alpha$-harmonic equations: $\Delta_{\alpha}(f(z))=\partial{z}[(1-{|{z}|}^{2})^{-\alpha} \overline{\partial}{z}f](z)=g(z)$, where…
We set new dual problems for the weighted spaces of holomorphic functions of one variable in domains on the complex plane, namely: nontriviallity of a given space, description of zero sets, description of (non-)uniqueness sets, the…
For analytic functions f(z) in the closed unit disk \bar{U}, two boundary points z_1 and z_2 such that \alpha = (f'(z_1)+f'(z_2))/2 in f'(U) are considered. The object of the present paper is to discuss some interesting conditions for f(z)…
We show that the Christoffel function (CF) factorizes (or can be disintegrated) as the product of two Christoffel functions, one associated with the marginal and the another related to the conditional distribution, in the spirit of "the CF…
The $\varphi$-order was introduced in 2009 for meromorphic functions in the unit disc, and was used as a growth indicator for solutions of linear differential equations. In this paper, the properties of meromorphic functions in the complex…
Suppose $Y$ is a regular covering of a graph $X$ with covering transformation group $\pi = \mathbb{Z}$. This paper gives an explicit formula for the $L^2$ zeta function of $Y$ and computes examples. When $\pi = \mathbb{Z}$, the $L^2$ zeta…
A connection between the zeta functions of zeros and poles of a meromorphic function has been established, and using it, a criterion for the absence of zeros has been derived. Sufficient conditions for the existence of zeros of sums of…
In this paper, we investigate meromorphic solutions in $\mathbb{C}^m$ of the nonlinear differential equation \[\displaystyle f^n\partial_u(f)g^n\partial_u(g)=1,\] where $\partial_u(f)=\sum_{j=1}^mu_j\partial_j(f)$ and $\sum_{j=1}^m u_j\neq…
In this paper we establish a close connection between three notions at- tached to a modular subgroup. Namely the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action…
In this paper, we give a definition of Eremenko's point of a meromorphic function with infinitely many poles and a condition for its existence in narrow annuli in terms of a covering theorem of annulus.
We define the class of Left Located Divisor (LLD) meromorphic functions and their vertical order $m_0(f)$ and their convergence exponent $d(f)$. When $m_0(f)\leq d(f)$ we prove that their Weierstrass genus is minimal. This explains the…