Entire Solutions for Some Fermat Type Functional Equations Concerning Difference and Partial Differential in ℂ2

2021 ◽  
Author(s):  
X.-M. Zheng ◽  
H.-Y. Xu
Keyword(s):  
Functional Equations ◽  
Entire Solutions ◽  
Partial Differential

Solutions for Several Quadratic Trinomial Difference Equations and Partial Differential Difference Equations in C2

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 126
Author(s):  
Hong Li ◽  
Hongyan Xu
Keyword(s):  
Positive Integer ◽  
Finite Order ◽  
Difference Equations ◽  
Functional Equations ◽  
Growth Order ◽  
Entire Solutions ◽  
Partial Differential ◽  
Integer Order ◽  
Significant Difference ◽  
Quadratic Trinomial

This article is to investigate the existence of entire solutions of several quadratic trinomial difference equations f(z+c)2+2αf(z)f(z+c)+f(z)2=eg(z), and the partial differential difference equations f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z12=eg(z),f(z+c)2+2αf(z+c)∂f(z)∂z1+∂f(z)∂z2+∂f(z)∂z1+∂f(z)∂z22=eg(z). We establish some theorems about the forms of the finite order transcendental entire solutions of these functional equations. We also list a series of examples to explain the existence of the finite order transcendental entire solutions of such equations. Meantime, some examples show that there exists a very significant difference with the previous literature on the growth order of the finite order transcendental entire solutions. Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order. These results make a few improvements of the previous theorems given by Xu and Cao, Liu and Yang.

scholarly journals Entire Solutions of

2005 ◽  
Vol 178 ◽  
pp. 151-162 ◽  
Author(s):  
Bao Qin li
Keyword(s):  
Entire Function ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Entire Solutions ◽  
Partial Differential

The paper is concerned with description of entire solutions of the partial differential equations where m ≥ 2, n ≥ 2 are integers and g is a polynomial or an entire function in C2. Descriptions are given and complemented by various examples.

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

scholarly journals A Linear Homogeneous Partial Differential Equation with Entire Solutions Represented by Laguerre Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Xin-Li Wang ◽  
Feng-Li Zhang ◽  
Pei-Chu Hu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Laguerre Polynomials ◽  
Convergent Series ◽  
Entire Solutions ◽  
Partial Differential ◽  
Order And Type

We study a homogeneous partial differential equation and get its entire solutions represented in convergent series of Laguerre polynomials. Moreover, the formulae of the order and type of the solutions are established.

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