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<p>By using asymptotic method, we verify the existence on the slowly growing solutions to second order difference equations discussed by Ishizaki-Yanagihara’s Wiman-Valiron method and Ishizaki-Wen’s binomial series method. The classical problem on finding conditions on the polynomial coefficients <i>P</i><sub><i>j</i></sub>(<i>z</i>) (<i>j</i> = 0, 1, 2) and <i>F</i>(<i>z</i>) to guarantee that all nontrivial solutions of complex second order difference equation <i>P</i><sub>2</sub>(<i>z</i>)<i>f</i>(<i>z</i> + 2) + <i>P</i><sub>1</sub>(<i>z</i>)<i>f</i>(<i>z</i> + 1) + <i>P</i><sub>0</sub>(<i>z</i>)<i>f</i>(<i>z</i>) = <i>F</i>(<i>z</i>) has slowly growing solutions with order 1/2 is detected.</p>

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Detecting the slowly growing solutions of second order linear difference equations

  • 期刊论文

  • 发表日期：

    2025年2月17日

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  • Zongxuan Chen,
  • Zhibo Huang,
  • Jun Wang,
  • Xiumin Zheng

摘要

By using asymptotic method, we verify the existence on the slowly growing solutions to second order difference equations discussed by Ishizaki-Yanagihara’s Wiman-Valiron method and Ishizaki-Wen’s binomial series method. The classical problem on finding conditions on the polynomial coefficients Pj(z) (j = 0, 1, 2) and F(z) to guarantee that all nontrivial solutions of complex second order difference equation P2(z)f(z + 2) + P1(z)f(z + 1) + P0(z)f(z) = F(z) has slowly growing solutions with order 1/2 is detected.