DOI: 10.1017/s0004972726102202 ISSN: 0004-9727
COMPLETE CLASSIFICATION OF MEROMORPHIC SOLUTIONS OF A FERMAT-TYPE DIFFERENCE EQUATION
OLEKSANDR VOLKOV Abstract
We investigate meromorphic solutions of the difference equation
f
(
z
)
2
+
f
(
z
+
c
)
=
h
(
z
)
,
c
∈
C
∖
{
0
}
,
$$ \begin{align*} f(z)^2+f(z+c)=h(z), \quad c\in\mathbb C\setminus\{0\}, \end{align*} $$
where
h
is a transcendental meromorphic function satisfying
N
(
r
,
h
)
+
N
(
r
,
1
/
h
)
=
S
(
r
,
h
)
.
$ N(r,h)+N(r,{1}/{h})=S(r,h). $
upper N left parenthesis r comma h right parenthesis plus upper N left parenthesis r comma 1 divided by h right parenthesis equals upper S left parenthesis r comma h right parenthesis period
We prove that every solution of hyper-order less than
1
$1$
1
is of the form
f
(
z
)
=
e
a
z
+
b
−
1
,
h
(
z
)
=
e
2
a
z
+
2
b
,
$$ \begin{align*} f(z)=e^{az+b}-1, \quad h(z)=e^{2az+2b}, \end{align*} $$
where