Abstract
Abstract This paper is devoted to exploring the existence and the forms of entire solutions of several quadratic trinomial differential difference equations with more general forms. Some results about the forms of entire solutions for these equations are some extensions and generalizations of the previous theorems given by Liu, Yang and Cao. We also give a series of examples to explain the existence of the finite order transcendental entire solutions of such equations.
Highlights
The main aim of this paper is to investigate the transcendental entire solutions with finite order of the quadratic trinomial difference equation f (z + c)2 + 2αf (z)f (z + c) + f (z)2 = eg(z), (1)
Gross [4] had discussed the existence of solutions of equation (3) and showed that the entire solutions are f = cosa(z), g = sina(z), where a(z) is an entire function
Motivated by the above question, this article is concerned with the entire solutions for the difference equation (1) and the differential difference equation (2)
Summary
They pointed out that the transcendental entire solution with finite order of the latter must satisfy f (z) = sin(z ± Bi), where B is a constant and c = 2kπ or c = (2k + 1)π, k is an integer. ), for example, f (z) eaz is a finite order entire solution of the first equation, if For α2 ≠ 0, 1, Liu and Yang [9] in 2016 studied the existence and the form of solutions of some quadratic trinomial functional equations and obtained the following results in equations (1) and (2).
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