### A note on algebraic differential equations whose coefficients are entire functions of finite order

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This paper is concerned with a second-order functional differential equation of the form ${x}^{\text{'}\text{'}}\left(z\right)=x(az+b{x}^{\text{'}}\left(z\right))$ with the distinctive feature that the argument of the unknown function depends on the state derivative. An existence theorem is established for analytic solutions and systematic methods for deriving explicit solutions are also given.

In this article we investigate the question [of] how meromorphic differential equations can be simplified by meromorphic equivalence. In the case of equations of block size 1, which generalizes the case of distinct eigenvalues, we identify a class of equations which are simplest possible in the sense that they carry the smallest number of parameters whithin their equivalence classes. We also discuss conditions under which individual equations can be simplified. Particular attention is paid to the...

We consider the Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions generated by elements of the symmetric group $${\mathcal{S}}_{n}$$ n. We assume that parameter ρ = ±1. In previous paper [5] we proved that the fundamental solution of the corresponding KZ-equation is rational. Now we construct this solution in the explicit form.

We study extension of $p$-trigonometric functions ${sin}_{p}$ and ${cos}_{p}$ to complex domain. For $p=4,6,8,\cdots $, the function ${sin}_{p}$ satisfies the initial value problem which is equivalent to (*) $$-{\left({u}^{\text{'}}\right)}^{p-2}{u}^{\text{'}\text{'}}-{u}^{p-1}=0,\phantom{\rule{1.0em}{0ex}}u\left(0\right)=0,\phantom{\rule{1.0em}{0ex}}{u}^{\text{'}}\left(0\right)=1$$ in $\mathbb{R}$. In our recent paper, Girg, Kotrla (2014), we showed that ${sin}_{p}\left(x\right)$ is a real analytic function for $p=4,6,8,\cdots $ on $(-{\pi}_{p}/2,{\pi}_{p}/2)$, where ${\pi}_{p}/2={\int}_{0}^{1}{(1-{s}^{p})}^{-1/p}$. This allows us to extend ${sin}_{p}$ to complex domain by its Maclaurin series convergent on the disc $\{z\in \u2102:|z|<{\pi}_{p}/2\}$. The question is whether this extensions ${sin}_{p}\left(z\right)$ satisfies (*) in the sense of differential equations in complex domain. This interesting...

We investigate the growth and fixed points of meromorphic solutions of higher order linear differential equations with meromorphic coefficients and their derivatives. Our results extend the previous results due to Peng and Chen.