/pdf/on-the-equation-f-n-g-n-1-rfnpxxrnei.pdf

On the equation $f^n + g^n = 1$

Publisher Summary This chapter discusses the Phragmen-Lindelof principle and a class of functional differential equations. The experimental mathematics and the computerhave played a role in motivating the work on this principle. One added fruitful result of using the computer is that many graphs were produced.Since graphs of solutions to retarded ordinary differential equations are rather rare, one include some of the more interesting oscillatory ones. The work was motivated by the results of systematic computer studies made of particular cases. These experiments suggested the likelihood of unbounded oscillatory solutions to equation.

The phragmén-lindelöf principle and a class of functional differential equations

We mainly discuss entire solutions with finite order of the following Fermat type differential-difference equations

 $$\begin{array}{ll}(f)^{n}+f(z+c)^{m}=1;\\f^{\prime}(z)^{n}+f(z+c)^{m}=1;\\ f^{\prime}(z)^{n}+[f(z+c)-f(z)]^{m}=1,\end{array}$$
 where m, n are positive integers.

Entire solutions of Fermat type differential-difference equations

In this paper, we will investigate the properties of entire solutions with finite order of the Fermat type difference or differential-difference equations. This is continuation of a recent paper (Liu et al. in Arch. Math. 99, 147–155, 2012). In addition, we also consider the value distribution and growth of the entire solutions of linear differential-difference equation $f^{(k)}(z)=h(z)f(z+c),$ where $h(z)$ is a non-zero meromorphic function, $c$ is a non-zero constant. Our results partially answer the question given in Liu et al. (Arch. Math. 99, 147–155, 2012).

Fred Gross

Papers

On the Functional Equation f n +g n = h n

On a Remark of Utz

Advanced Problems: 5300-5309

On prime periodic entire functions

On Functional Values of Families of Analytic Functions