Exponential polynomials as solutions of certain nonlinear difference equations
Exponential polynomials as solutions of certain nonlinear difference equations
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DOI：
10.1007/s10114-012-1484-2
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发表时间：
2012-01
期刊：
影响因子：
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通讯作者：
Z. Wen;J. Heittokangas;Ilpo Lain
中科院分区：
文献类型：
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作者：
Z. Wen;J. Heittokangas;Ilpo Lain
最近,c . c。Yang和I. Laine研究了形式为fn+L(z, f) =h的非线性微分-差分方程的有限阶全解，其中≥2为整数。特别地，已知方程f(z)2+q(z)f(z+1) =p(z)，其中ep(z),q(z)是多项式，没有有限阶的超越全解。假设q(z)也是一个多项式，且c∈，则形式为f(z)n+q(z)eQ(z)f(z+c) =p(z)的方程确实具有有限阶全解。将根据增长和零分布对这些解进行分类。特别地，证明了任何指数多项式解必须约简成一个相当特定的形式。这个推理依赖于N. Steinmetz早先的一篇论文。
Recently, C.-.C. Yang and I. Laine have investigated finite order entire solutionsfof nonlinear differential-difference equations of the formfn+L(z, f) =h, wheren≥ 2 is an integer. In particular, it is known that the equationf(z)2+q(z)f(z+1) =p(z), wherep(z),q(z) are polynomials, has no transcendental entire solutions of finite order. Assuming thatQ(z) is also a polynomial andc∈ ℂ, equations of the formf(z)n+q(z)eQ(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.