两类Cauchy数的概率表示及应用
Probabilistic Representations and Applications of Two Kinds of Cauchy Numbers
DOI: 10.12677/AAM.2020.97117, PDF, 下载: 570  浏览: 752  科研立项经费支持
作者: 常桂松, 徐 晨：东北大学数学系，辽宁 沈阳
关键词: Cauchy数；均匀分布；阶乘矩；Cauchy Numbers； Uniform Distribution； Factorial Moment
摘要: 本文利用(0,1)区间上的均匀分布的阶乘矩，提出了第一类Cauchy数和第二类Cauchy数的概率表示。利用概率论的方法和技巧，证明了包含第一类Cauchy数和第二类Cauchy数的一些递推关系和一部分有趣的恒等式。最后证明了第一类Cauchy数和第二类Cauchy数的卷积公式。
Abstract: In this paper, we prove that the Cauchy numbers of the first and the second kind both are factorial moments of uniform random variable, and make use of the probabilistic methods and skills to the computation of combinatorial sums. As a result of the applications, some convolutions of the Cauchy numbers are obtained.
文章引用：常桂松, 徐晨. 两类Cauchy数的概率表示及应用[J]. 应用数学进展, 2020, 9(7): 987-991. https://doi.org/10.12677/AAM.2020.97117

参考文献

[1] Comtet, L. (1974) Advanced Combinatorics. D. Reidel Publishing Company, Dordrecht.
[2] Merlini, D., Sprugnoli, R. and Verri, M.C. (2006) The Cauchy Numbers. Discrete Mathematics, 306, 1906-1920.
https://doi.org/10.1016/j.disc.2006.03.065
[3] Zhao, F.Z. (2009) Sums of Products of Cauchy Numbers. Discrete Mathematics, 309, 3830-3842.
https://doi.org/10.1016/j.disc.2008.10.013
[4] Komatsu, T. (2013) Sums of Products of Cauchy Numbers, Including Poly-Cauchy Numbers. Journal of Discrete Mathematics, 2013, Article ID: 373927, 10 p.
https://doi.org/10.1155/2013/373927
[5] 郑德印. 两类Cauchy数的共同推广[J]. 大连理工大学学报, 2004, 44(4): 66-69.
[6] 李志荣, 肖冰, 袁文俊. 广义Cauchy数的一些恒等式[J]. 吉林大学学报(理学版), 2009, 47(1): 17-20.

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