Uniqueness of exponential polynomials

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.