封闭式公式、行列式表达、递推关系、幂级数和克拉克-伊斯梅尔两个猜想中使用的几个函数的特殊值

IF 4.6 2区数学 Q1 MATHEMATICS, APPLIED

Applied and Computational Mathematics Pub Date : 2023-10-19 DOI:10.30546/1683-6154.22.4.2023.443

Yan-Fang Li, D. Lim, Feng Qi

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引用次数: 0

摘要

在这篇论文中，作者通过 Fa\`a di Bruno 的著名公式，借助贝尔局部多项式的几个等式，通过两个可微分函数之比导数的公式，并利用其他技术、作者建立了伯努利数和第二类斯特林数的闭式公式，提出了行列式表达式，推导了递推关系，获得了幂级数，并计算了函数 $/frac{v^j}{1-\operatorname{e}^{-v}}$、其导数以及克拉克-伊斯梅尔的两个猜想中使用的相关导数的特殊值。通过这些结果，作者还发现了海森堡矩阵的行列式公式，并推导出与函数及其导数相关的序列的对数凸性。

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Closed-form formulas, determinantal expressions, recursive relations, power series, and special values of several functions used in Clark–İsmail’s two conjectures

In the paper, by virtue of the famous formula of Fa\`a di Bruno, with the aid of several identities of partial Bell polynomials, by means of a formula for derivatives of the ratio of two differentiable functions, and with availability of other techniques, the authors establish closed-form formulas in terms of the Bernoulli numbers and the second kind Stirling numbers, present determinantal expressions, derive recursive relations, obtain power series, and compute special values of the function $\frac{v^j}{1-\operatorname{e}^{-v}}$, its derivatives, and related ones used in Clark--Ismail's two conjectures. By these results, the authors also discover a formula for the determinant of a Hessenberg matrix and derive logarithmic convexity of a sequence related to the function and its derivatives.

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来源期刊

Applied and Computational Mathematics 数学-应用数学

CiteScore

8.80

自引率

5.00%

发文量

审稿时长

6 months

期刊介绍： Applied and Computational Mathematics (ISSN Online: 2328-5613, ISSN Print: 2328-5605) is a prestigious journal that focuses on the field of applied and computational mathematics. It is driven by the computational revolution and places a strong emphasis on innovative applied mathematics with potential for real-world applicability and practicality. The journal caters to a broad audience of applied mathematicians and scientists who are interested in the advancement of mathematical principles and practical aspects of computational mathematics. Researchers from various disciplines can benefit from the diverse range of topics covered in ACM. To ensure the publication of high-quality content, all research articles undergo a rigorous peer review process. This process includes an initial screening by the editors and anonymous evaluation by expert reviewers. This guarantees that only the most valuable and accurate research is published in ACM.