相关论文: All possible orders less than 1 of transcendental …
We show a necessary and sufficient condition on the existence of finite order entire solutions of linear differential equations $$ f^{(n)}+a_{n-1}f^{(n-1)}+\cdots+a_1f'+a_0f=0,\eqno(+) $$ where $a_i$ are exponential sums for…
We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
In this paper, we study the uniqueness of the differential-difference polynomials of entire functions on $\mathbb{C}^{n}$. We prove the following result: Let $f(z)$ be a transcendental entire function on $\mathbb{C}^{n}$ of hyper-order less…
Assuming that $A_0,\ldots,A_{n-1}$ are entire functions and that $p\in \{0,\ldots,n-1\}$ is the smallest index such that $A_p$ is transcendental, then, by a classical theorem of Frei, each solution base of the differential equation…
Let $f$ be a transcendental entire function with hyper-order strictly less than 1 and having a Borel exceptional small function. If $f$ and $\Delta^n f$, or $f'$ and $f(z+1)$, share a function CM, then the exact form of $f$ is determined,…
We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1…
In this paper, we mainly investigate on the finite order transcendental entire solutions of two Fermat types delay-differential and one Fermat type c-shift equations, as these types were not considered earlier. Our results improve those of…
We study higher order linear differential equation $y^{(k)}+A_1(z)y=0$ with $k\geq2$, where $A_1=A+h$, $A$ is a transcendental entire function of finite order with $\frac{1}{2}\leq \mu(A)<1$ and $h\neq0$ is an entire function with…
Considering differential equation f''+A(z)f'+B(z)f=0, where A(z) and B(z) are entire complex functions, our results revolve around proving all non-trivial solutions are of infinite order taking various restrictions on coefficients A(z) and…
In this paper we prove that the PDE $p(D)f=q,$ where $p$ and $q$ are multivariate polynomials, has a solution in the space of polynomials of total degree not exceeding ${n+s},$ where $n$ is the degree of $q$ and $s$ is the zero order of…
In this study, we show that all non-trivial solutions of $f"+A(z)f'+B(z)f=0$ have infinite order, provided that the entire coefficient $A(z)$ has certain restrictions and $B(z)$ has multiply-connected Fatou component. We also extend these…
We prove some new results on existence of solutions to first--order ordinary differential equations with deviating arguments. Delay differential equations are included in our general framework, which even allows deviations to depend on the…
In this paper we mainly study the existence and the form of entire solutions with finite order for the following system of Fermat-type difference and partial differential-difference equations $$\begin{cases} f_1(z)^2+(\Delta_cf_2(z))^2=1\cr…
In this short paper we identify special systems of (an arbitrary number) N of first-order Difference Equations with nonlinear homogeneous polynomials of arbitrary degree M in their right-hand sides, which feature very simple explicit…
In this paper, we will prove that all non-trivial solutions of $f''+A(z)f'+B(z)f=0$ are of infinite order, where we have some restrictions on entire functions $A(z)$ and $B(z)$.
In this article, we study the growth of solutions of the homogeneous complex linear differential equation \begin{equation*} f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f^{\prime}+ A_{0}(z)f=0, \end{equation*}% where the coefficients…
We show that the higher order linear differential equation possesses all solutions of infinite order under certain conditions by extending the work of authors about second order differential equation \cite{dsm2}.
We consider transcendental entire functions of finite order for which the zeros and $1$-points are in disjoint sectors. Under suitable hypotheses on the sizes of these sectors we show that such functions must have a specific form, or that…
Given a subset $S=\{s_0, s_1\}$ of the complex plane with two points and an infinite subset ${\mathscr S}$ of $S\times {\mathbb N}$, where ${\mathbb N}=\{0,1,2,\dots\}$ is the set of nonnegative integers, we ask for a lower bound for the…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…