Abstract
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.
Highlights
Let α2 (6= 0, 1), z + c = (z1 + c1, z2 + c2 ) for z = (z1, z2 ) and c = (c1, c2 )
Our results show that some functional equations can admit the transcendental entire solutions with any positive integer order
This paper is devoted to investigating the transcendental entire solutions with finite order of the quadratic trinomial difference equation
Summary
G. Saleeby [20,21] studied the forms of the entire and meromorphic solutions of some partial differential equations, extended some of the above conclusions to the case of several complex variables, and obtained the following results. Where g is a polynomial or an entire function in C2 and obtained some results on the forms of entire solution of Equation (11) as follows: Theorem 3 ([22], Theorem 2.1). G. Saleeby [23] further investigated the entire and meromorphic solutions for the quadratic trinomial functional equations f 2 + 2α f g + g2 = 1, α2 6= 1, α ∈ C,. What would happen to the existence and form of solution of Equation (12) when g is replaced of some special forms of f , and the right side of those equations 1 is replaced by a function e φ in Theorem D, where φ is a polynomial?
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