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<p>In the paper, we find out the precise form of the finite order entire solutions of the following difference equation <Equation ID="Equ59"> <EquationSource Format="TEX">\(\begin{aligned} f(z)=\mathop {{\mathop {\sum }\nolimits ^n_{j=0}}}\limits a_j f(z+jc), \end{aligned}\)</EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(a_0, a_1,\ldots ,a_n(\ne 0)\in \mathbb {C}\)</EquationSource> </InlineEquation>. Also in the paper we study the difference analogue of Brück conjecture and derive a uniqueness result of finite order entire function <i>f</i>(<i>z</i>) having a Borel exceptional small function of <i>f</i>(<i>z</i>), when <i>f</i>(<i>z</i>) and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathop {{\mathop {\sum }\nolimits ^n_{j=0}}}\limits a_j f(z+jc)\)</EquationSource> </InlineEquation> share a small function of <i>f</i>(<i>z</i>). The obtained results, significantly generalize and improve the recent results of Chen and Chen (Taiwanese J Math 18(3):711–729, 2014), Liao and Zhang (Bull Korean Math Soc 53(1):49–60, 2016), Lü et al. (Results Math 74(30):1–18, 2019) and Huang and Zhang (Anal Math 44:461–473, 2018) in a large scale. Some examples are given to ensure the necessity of the condition (s) of our main results.</p>

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Entire solutions of a certain type of difference equation and the difference analogue of Brück conjecture

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Sujoy Majumder,
  • Debabrata Pramanik,
  • Pradip Das

摘要

In the paper, we find out the precise form of the finite order entire solutions of the following difference equation \(\begin{aligned} f(z)=\mathop {{\mathop {\sum }\nolimits ^n_{j=0}}}\limits a_j f(z+jc), \end{aligned}\) where \(a_0, a_1,\ldots ,a_n(\ne 0)\in \mathbb {C}\) . Also in the paper we study the difference analogue of Brück conjecture and derive a uniqueness result of finite order entire function f(z) having a Borel exceptional small function of f(z), when f(z) and \(\mathop {{\mathop {\sum }\nolimits ^n_{j=0}}}\limits a_j f(z+jc)\) share a small function of f(z). The obtained results, significantly generalize and improve the recent results of Chen and Chen (Taiwanese J Math 18(3):711–729, 2014), Liao and Zhang (Bull Korean Math Soc 53(1):49–60, 2016), Lü et al. (Results Math 74(30):1–18, 2019) and Huang and Zhang (Anal Math 44:461–473, 2018) in a large scale. Some examples are given to ensure the necessity of the condition (s) of our main results.