Exponential Polynomials as Solutions of Certain Nonlinear Difference Equations

• Published in: Acta mathematica Sinica. English series Vol. 28; no. 7; pp. 1295 - 1306
• Main Authors: Wen, Zhi Tao, Heittokangas, Janne, Lain, Ilpo
• Format: Journal Article
• Language: English
• Published: Heidelberg Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.07.2012; Springer Nature B.V
• Subjects: Classification; Difference equations; Differential equations; Integers; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinearity; Polynomials; Reasoning; Studies; Theorems; 函数; 差分方程解; 微分差分方程; 指数多项式; 整数; 欧共体; 零分布; 非线性; difference equation; Convex hull; 39B32; exponential polynomial; Nevanlinna theory; 30D35; entire solution
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-012-1484-2

Recently, C.-C. Yang and L Laine have investigated finite order entire solutions f of non- linear differential-difference equations of the form f^n ＋ L（z, f） -= h, where n ≥ 2 is an integer. In particular, it is known that the equation f（z）^2 ＋ q（z）f（z ＋ 1） = p（z）, where p（z）, q（z） are polynomials, has no transcendental entire solutions of finite order. Assuming that Q（z） is also a polynomial and c E C, equations of the form f（z）^n ＋ q（z）e^Q（Z）f（z ＋ c） = p（z） do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.