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基于概率统计方法证明若干组合恒等式
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Abstract:
组合恒等式的发现与证明一直是组合数学的一个主要分支，一些组合等式因其复杂性难以直接证明，如何给出组合恒等式的简洁证明是组合数学的重要研究方向。本文应用负二项分布卷积的不同表达形式，与负二项分布的可加性等性质，发现并证明了若干组合恒等式。
Finding and proving combinatorial identities is an important part of combinatorial mathematics. However, some combinatorial identities contain computational complexity, which hinders the direct proofs. So it is an important research direction in combinatorial mathematics to give concise proofs of combinatorial identities. In this paper, some combinatorial identities are found and proved by using different expressions of convolution of negative binomial distribution and the additivity of negative binomial distribution.
| [1] | Vellaisamy, P. and Zeleke, A. (2019) Probabilistic Proofs of Some Beta-Function Identities. Journal of Integer Sequenc-es, 22, 1-10. |
| [2] | Ge, H.P. and Zhao, X.Q. (2010) Some Applications of Probabilistic Methods in Combinatorial Mathematics. Periodical of Ocean University of China, 9, 230-234. |
| [3] | Balakrishnan, N. (2012) Advances in Combi-natorial Methods and Applications to Probability and Statistics. Birkh？user, Boston, 22-96. |
| [4] | Xu, C. and Chang, G.S. (2017) Exact Distribution of the Convolution of Negative Binomial Random Variables. Communications in Statis-tics—Theory and Methods, 46, 2851-2856. https://doi.org/10.1080/03610926.2015.1053931 |
| [5] | Vellaisamy, P. and Upadhye, N.S. (2009) On the Sums of Compound Negative Binomial and Gamma Random Variables. Journal of Applied Probability, 46, 272-283. https://doi.org/10.1239/jap/1238592129 |
| [6] | Furman, E. (2006) On the Convo-lution of the Negative Binomial Random Variables. Statistics and Probability Letters, 77, 169-172. https://doi.org/10.1016/j.spl.2006.06.007 |
| [7] | Zhao, F.Z. (2016) Some Results for the Inverse Moment of the n-Fold Convolution of the Zero-Truncated Negative Binomial Distribution. The Journal of the Indian Mathematical Soci-ety, 83, 199-208. |
| [8] | Imoto, T. (2015) Convolution of Binomial and Negative Binomial Variables. Communications in Statistics—Theory and Methods, 44, 5005-5022. https://doi.org/10.1080/03610926.2013.809110 |
| [9] | Ahuja, J.C. and Enneking, E.A. (1974) Convolution of Independent Left-Truncated Negative Binomial Variables and Limiting Dis-tributions. Annals of the Institute of Statistical Mathematics, 26, 265-270.
https://doi.org/10.1007/BF02479821 |
| [10] | Sen, A. and Balakrishnan, N. (1999) Convolution of Geometrics and a Reliability Problem. Statistics and Probability Letters, 43, 421-426. https://doi.org/10.1016/S0167-7152(98)00284-3 |
| [11] | Psarrakos, G. (2009) A Note on Convolutions of Compound Geometric Distributions. Statistics and Probability Letters, 79, 1231-1237. https://doi.org/10.1016/j.spl.2009.01.012 |
| [12] | Boland, P.J., El-Neweihi, E. and Proschan, F. (1994) Schur Proper-ties of Convolutions of Exponential and Geometric Random Variables. Journal of Multivariate Analysis, 48, 157-167. https://doi.org/10.1016/0047-259X(94)80009-K |
| [13] | Comtet, L. 著. 高等组合学: 有限和无限展开的艺术[M]. 谭明术, 郝培峰, 杨利民, 张玉森, 唐朝平, 译. 大连:大连理工大学出版社, 1991: 151-160. |
| [14] | Sun, P. (2007) Moment Representation of Bernoulli Polynomial, Euler Polynomial and Gegenbauer Polynomials. Statistics and Proba-bility Letters, 77, 748-751. https://doi.org/10.1016/j.spl.2006.11.011 |
| [15] | 祁兰, 张媛. 关于Bell多项式的一些恒等式[J]. 内蒙古师范大学学报: 自然科学汉文版, 2020, 49(2): 5. |
| [16] | 杨继真, 王云鹏. 一类包含有完全Bell多项式的恒等式(英文) [J]. 数学季刊(英文版), 2017, 32(4): 89-98. |
| [17] | 过静, 李小雪. 关于Bell多项式及其它的一些恒等式[J]. 数学杂志, 2017, 37(6): 1201-1206. |
| [18] | 道如娜图亚, 乌云高娃. 关于第三类退化的Poly-Cauchy多项式的组合恒等式[J]. 应用数学进展, 2022, 11(1): 492-502. https://doi.org/10.12677/aam.2022.111057 |
