Uniqueness of exponential polynomials

Abstract

In this article, we study the uniqueness of exponential polynomials and mainly prove: Let nn be a positive integer, let pi(z)(i=1,2,…,n){p}_{i}\left(z)\hspace{0.33em}\left(i=1,2,\ldots ,n) be nonzero polynomials, and let ci≠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)=p1(z)ec1z+p2(z)ec2z+⋯+pn(z)ecnzg\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 aa and bb CM (counting multiplicities), where aa and bb are two distinct finite complex numbers, then one of the following cases must occur: (i)n=1n=1.If a≠0a\ne 0, b=0b=0, then either f(z)≡g(z)f\left(z)\equiv g\left(z) or f(z)g(z)≡a2f\left(z)g\left(z)\equiv {a}^{2};If a=0a=0, b≠0b\ne 0, then either f(z)≡g(z)f\left(z)\equiv g\left(z) or f(z)g(z)≡b2f\left(z)g\left(z)\equiv {b}^{2};If a≠0a\ne 0, b≠0b\ne 0, then either f(z)≡g(z)f\left(z)\equiv g\left(z) or f(z)g(z)≡(a+b)g(z)−abf\left(z)g\left(z)\equiv \left(a+b)g\left(z)-ab.(ii)n≥2n\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.

Keywords

• exponential polynomials
• uniqueness
• differences
• 30d35