Affiliation:
1. Department of Mathematics, Hangzhou Dianzi University , Hangzhou 310018 , China
Abstract
Abstract
In this article, we study the uniqueness of exponential polynomials and mainly prove: Let
n
n
be a positive integer, let
p
i
(
z
)
(
i
=
1
,
2
,
…
,
n
)
{p}_{i}\left(z)\hspace{0.33em}\left(i=1,2,\ldots ,n)
be nonzero polynomials, and let
c
i
≠
0
(
i
=
1
,
2
,
…
,
n
)
{c}_{i}\ne 0\hspace{0.33em}\left(i=1,2,\ldots ,n)
be distinct finite complex numbers. Suppose that
f
(
z
)
f\left(z)
is an entire function,
g
(
z
)
=
p
1
(
z
)
e
c
1
z
+
p
2
(
z
)
e
c
2
z
+
⋯
+
p
n
(
z
)
e
c
n
z
g\left(z)={p}_{1}\left(z){e}^{{c}_{1}z}+{p}_{2}\left(z){e}^{{c}_{2}z}+\cdots +{p}_{n}\left(z){e}^{{c}_{n}z}
. If
f
(
z
)
f\left(z)
and
g
(
z
)
g\left(z)
share
a
a
and
b
b
CM (counting multiplicities), where
a
a
and
b
b
are two distinct finite complex numbers, then one of the following cases must occur:
(i)
n
=
1
n=1
.
If
a
≠
0
a\ne 0
,
b
=
0
b=0
, then either
f
(
z
)
≡
g
(
z
)
f\left(z)\equiv g\left(z)
or
f
(
z
)
g
(
z
)
≡
a
2
f\left(z)g\left(z)\equiv {a}^{2}
;
If
a
=
0
a=0
,
b
≠
0
b\ne 0
, then either
f
(
z
)
≡
g
(
z
)
f\left(z)\equiv g\left(z)
or
f
(
z
)
g
(
z
)
≡
b
2
f\left(z)g\left(z)\equiv {b}^{2}
;
If
a
≠
0
a\ne 0
,
b
≠
0
b\ne 0
, then either
f
(
z
)
≡
g
(
z
)
f\left(z)\equiv g\left(z)
or
f
(
z
)
g
(
z
)
≡
(
a
+
b
)
g
(
z
)
−
a
b
f\left(z)g\left(z)\equiv \left(a+b)g\left(z)-ab
.
(ii)
n
≥
2
n\ge 2
,
f
(
z
)
≡
g
(
z
)
f\left(z)\equiv g\left(z)
.
This is an extension of the result obtained in an earlier study on meromorphic functions in 1974.