Author:
Zhang Yueyang,Korhonen Risto
Abstract
AbstractIt is shown that if the equation $$\begin{aligned} f(z+1)^n=R(z,f), \end{aligned}$$
f
(
z
+
1
)
n
=
R
(
z
,
f
)
,
where R(z, f) is rational in both arguments and $$\deg _f(R(z,f))\not =n$$
deg
f
(
R
(
z
,
f
)
)
≠
n
, has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term R(z, f) takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobian elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case $$\deg _f(R(z,f))=n$$
deg
f
(
R
(
z
,
f
)
)
=
n
of the equation above and thus provide a complete difference analogue of Steinmetz’ generalization of Malmquist’s theorem.
Funder
University of Eastern Finland (UEF) including Kuopio University Hospital
Publisher
Springer Science and Business Media LLC
Subject
Computational Mathematics,General Mathematics,Analysis
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