摘要
[目的]研究一类费马型差分方程f^(n)(z)+P(z)f m(z+c)=H(z)亚纯解的性质,其中H(z)=H_(1)(z)e^(μ)1^(z^(q))+…+H_(k)(z)e^(μ)k^(z^(q)),q≥1,k≥1为整数,m和n为正整数,c为非0复常数,μ_(1),…,μ_(k)是不同的非0复数,P(z),H_(1)(z),…,H_(k)(z)都是级小于q的整函数;探索更广泛的参数范围下解的存在性.[方法]运用Nevanlinna理论与Cartan第二基本定理,结合差分对数导数引理、Borel型定理等工具,对解的级、零点分布及函数结构进行精细估计.通过引入典型乘积和线性无关性讨论,将问题转化为整函数与指数函数的组合分析.[结果]在m≥k,n>2k或者m<k,n>k+m的条件下,获得了方程解存在的必要条件与具体表达式.当k=1时,得到n=m以及解的显式形式,即解为单指数型函数;当k=2时,解为指数型函数并满足某些特定比例关系.给出了相应的例子证明了解的存在性,同时,构造了反例证明条件的必要性,并给出了一系列关于解不存在性的判据.该结果揭示了方程右端指数项个数与方程解的结构直接的紧密联系,是已有结果的广泛形式.[结论]本文结果减弱了已有定理中对n和m关系的限制,扩展了费马型差分方程可解的参数范围,揭示了右端项指数个数与解的结构之间的深刻联系.特别地,本文证明只有在特定的指数匹配条件下,此类方程才可能具有显式闭形解.所采用方法具有一般性,可为同类差分方程的研究提供新思路.
[Objective]Properties of meromorphic solutions for the Fermat-type difference equation f^(n)(z)+P(z)f m(z+c)=H(z)are investigated,where H(z)=H_(1)(z)e^(μ)1^(z^(q))+…+H_(k)(z)e^(μ)k^(z^(q));q≥1,k≥1,are integers;m and n are positive integers;c is a nonzero complex constant;μ_(1),…,μ_(k)are distinct nonzero complex numbers;and P(z),H_(1)(z),…,H_(k)(z)are entire functions of order less than q.The existence of solutions is investigated across a broad range of parameters.[Methods]Methods employed herein include Nevanlinna theory and Cartan's second fundamental theorem,combined with tools such as the difference logarithmic derivative lemma and Borel-type theorems.These techniques are utilized to conduct fine-grained estimates on the order,zero distribution,and functional structure of solutions.By introducing canonical products and analyzing linear independence,the problem is transformed into combinatorial analyses of entire functions and exponential functions.[Results]Under conditions m≥k,n>2k,or m<k,n>k+m,necessary conditions for the existence of solutions and their explicit expressions are obtained.When k=1,the case n=m is obtained along with the explicit form of the solution,which is a single exponential function.When k=2,the solution takes the form of exponential functions that satisfy certain specific proportional relationships.Corresponding examples are provided to demonstrate the existence of solutions.Meanwhile,counterexamples are constructed to prove the necessity of these conditions,and a series of criteria for the nonexistence of solutions are presented.These results reveal the close relationship between the number of exponential terms on the right-hand side of the equation and the structure of its solutions,representing a broad generalization of existing results.[Conclusions]Results of this work relax restrictions on the relationship between n and m in existing theorems,thereby expanding the parameter range,for which Fermat-type difference equations are solvable.They reveal a deep connection between the number of exponential terms on the right-hand side and the structure of these solutions.In particular,it is proven that explicit closed-form solutions are possible only under specific index-matching conditions.Finally,the adopted method is general and can provide new insights for the study of similar difference equations.
作者
夏梦婷
刘建明
商雨知
彭长文
XIA Mengting;LIU Jianming;SHANG Yuzhi;PENG Changwen(School of Mathematical Sciences,Guizhou Normal University,Guiyang 550025,China;School of Science,Kaili University,Kaili 556011,China;School of Science,China University of Mining and Technology,Beijing 100083,China;School of Science,Guiyang University,Guiyang 550005,China)
出处
《厦门大学学报(自然科学版)》
北大核心
2026年第2期361-367,共7页
Journal of Xiamen University:Natural Science
基金
国家自然科学基金(11861023)
贵阳学院博士科研启动项目(GYU-KY-[2025])。
