几类组合和式的相关同余式
Congruences of Several Kinds of Combinatorics Sums
摘要: 利用常数项方法，得到了几个包含常见组合序列如中心二项式系数、Catalan数、Motzkin数的部分和式的新的同余式，结合扩展的Zeilberger算法，讨论了一类含有中心二项式系数和式的同余性质之间的关系并提出了一个有待研究的问题。
Abstract: By using the method of constant term, we prove several congruences of combinatorial sums in-volving some usual combinatorial sequences such as central binomial coefficients, Catalan numbers and Motzkin numbers. Moreover, with the help of the extended Zeilberger algorithm, we explore the congruence relations of a certain kind of combinatorial sums involving central binomial coefficients and propose a valuable research problem.
文章引用：王明会, 靳海涛. 几类组合和式的相关同余式[J]. 理论数学, 2023, 13(4): 886-894. https://doi.org/10.12677/PM.2023.134094

参考文献

[1] Pan. H. and Sun, Z.W. (2006) Acombinatorial Identity with Applications to Catalan Numbers. Discrete Mathematics, 306, 1921-1940. [Google Scholar] [CrossRef] 
[2] 曹植坚. 涉及Apery多项式与中心二项式系数的同余式[D]: [硕士学位论文]. 南京: 南京大学, 2020.
[3] Apagodu, M. (2018) Elementary Proof of Congruences Involving Sum of Binomial Coefficients. International Journal of Number Theory, 14, 1547-1557. [Google Scholar] [CrossRef] 
[4] Liu, J.C. (2022) Supercongruences Involving Motzkin Numbers and Central Trinomial Coefficients. arXiv: 2208.10275
[5] Sun, Z.W. (2022) On Motzkin Numbers and Central Trinomial Coefficients. Advances in Applied Mathematics, 136, 102319. [Google Scholar] [CrossRef] 
[6] 孙智伟. 数论与组合中的新猜想[M]. 哈尔滨: 哈尔滨工业大学, 2021.
[7] Ahlgren, S. and Ono, K. (2000) A Gaussian Hypergeormetric Series Evaluation and Apéry Number Congruences. Journal fur die Reine und Angewandte Mathematik, 518, 187-212. [Google Scholar] [CrossRef] 
[8] Zudilin, W. (2009) Ramanujan-Type Supercongruences. Journal of Number Theory, 129, 1848-1857. [Google Scholar] [CrossRef] 
[9] Beukers, F. (1987) Another Congruence for the Apéry Numbers. Journal of Number Theory, 25, 201-210. [Google Scholar] [CrossRef] 
[10] Chen, W.Y.C., Hou, Q.H. and Zeilberger, D. (2015) Auto-mated Discovery and Proof of Congruence Theorems for Partial Sums of Combinatorial Sequences. Journal of Differ-ence Equations and Applications, 22, 780-788. [Google Scholar] [CrossRef] 
[11] Hou, Q.H., Mu, Y.P. and Zeilberger, D. (2021) Polynomial Reduction and Supercongruences. Journal of Symbolic Computation, 103, 127-140. [Google Scholar] [CrossRef] 
[12] Sun, Z.W. (2014) Congruences Involving Generalized Central Trinomial Coefficients. Science China Mathematics, 57, 1375-1400. [Google Scholar] [CrossRef] 
[13] Apagodu, M. and Zeilberger, D. (2017) Using the “Freshman’s Dream” to Prove Combinatorial Congruences. The American Mathematical Monthly, 124, 597-608. [Google Scholar] [CrossRef] 
[14] Apagodu, M. and Zeilberger, D. (2006) Multi-Variable Zeilberger and Almkvist-Zeilberger Algorithms and the Sharpening of Wilf-Zeilberger Theory. Advances in Applied Mathematics, 37, 139-152. [Google Scholar] [CrossRef] 

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