{"title":"价值共享和斯特林数字","authors":"Aimo Hinkkanen, Ilpo Laine","doi":"10.1007/s40315-024-00552-5","DOIUrl":null,"url":null,"abstract":"<p>Let <i>f</i> be an entire function and <i>L</i>(<i>f</i>) a linear differential polynomial in <i>f</i> with constant coefficients. Suppose that <i>f</i>, <span>\\(f'\\)</span>, and <i>L</i>(<i>f</i>) share a meromorphic function <span>\\(\\alpha (z)\\)</span> that is a small function with respect to <i>f</i>. A characterization of the possibilities that may arise was recently obtained by Lahiri. However, one case leaves open many possibilities. We show that this case has more structure than might have been expected, and that a more detailed study of this case involves, among other things, Stirling numbers of the first and second kinds. We prove that the function <span>\\(\\alpha \\)</span> must satisfy a linear homogeneous differential equation with specific coefficients involving only three free parameters, and then <i>f</i> can be obtained from each solution. Examples suggest that only rarely do single-valued solutions <span>\\(\\alpha (z)\\)</span> exist, and even then they are not always small functions for <i>f</i>.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Value Sharing and Stirling Numbers\",\"authors\":\"Aimo Hinkkanen, Ilpo Laine\",\"doi\":\"10.1007/s40315-024-00552-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <i>f</i> be an entire function and <i>L</i>(<i>f</i>) a linear differential polynomial in <i>f</i> with constant coefficients. Suppose that <i>f</i>, <span>\\\\(f'\\\\)</span>, and <i>L</i>(<i>f</i>) share a meromorphic function <span>\\\\(\\\\alpha (z)\\\\)</span> that is a small function with respect to <i>f</i>. A characterization of the possibilities that may arise was recently obtained by Lahiri. However, one case leaves open many possibilities. We show that this case has more structure than might have been expected, and that a more detailed study of this case involves, among other things, Stirling numbers of the first and second kinds. We prove that the function <span>\\\\(\\\\alpha \\\\)</span> must satisfy a linear homogeneous differential equation with specific coefficients involving only three free parameters, and then <i>f</i> can be obtained from each solution. Examples suggest that only rarely do single-valued solutions <span>\\\\(\\\\alpha (z)\\\\)</span> exist, and even then they are not always small functions for <i>f</i>.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-06-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s40315-024-00552-5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s40315-024-00552-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
假设 f 是一次函数,L(f) 是 f 的线性微分多项式,且系数不变。假设 f、\(f'\)和 L(f)共享一个关于 f 的微函数 \(\α(z)\)。然而,有一种情况留下了许多可能性。我们证明,这种情况的结构比预想的要复杂得多,而且对这种情况的更详细的研究还涉及第一和第二种斯特林数。我们证明,函数 \(\α \) 必须满足一个线性均质微分方程,其特定系数只涉及三个自由参数,然后可以从每个解中得到 f。例子表明,单值解 \(\alpha (z)\) 只在极少数情况下存在,即便如此,它们也并不总是 f 的小函数。
Let f be an entire function and L(f) a linear differential polynomial in f with constant coefficients. Suppose that f, \(f'\), and L(f) share a meromorphic function \(\alpha (z)\) that is a small function with respect to f. A characterization of the possibilities that may arise was recently obtained by Lahiri. However, one case leaves open many possibilities. We show that this case has more structure than might have been expected, and that a more detailed study of this case involves, among other things, Stirling numbers of the first and second kinds. We prove that the function \(\alpha \) must satisfy a linear homogeneous differential equation with specific coefficients involving only three free parameters, and then f can be obtained from each solution. Examples suggest that only rarely do single-valued solutions \(\alpha (z)\) exist, and even then they are not always small functions for f.