用“新生梦”证明组合同余式
Using the "Freshman's Dream" to Prove Combinatorial Congruences
摘要：
在最近一篇漂亮但技术性的文章中，William Y.C. Chen，Qing-Hu Hou和Doron Zeilberger开发了一种算法，用于寻找和证明许多组合序列的不定和的同余恒等式（模素数），即那些（如Catalan和Motzkin序列）可以用Laurent多项式的幂的常数项表示的序列。我们首先给出一个悠闲的阐述他们的基本但辉煌的方法，然后在两个方向上扩展它。洛朗多项式可以是多个变量，而不是单一的总和，我们有多个总和。事实上，我们甚至把这两个概括结合起来！最后，我们将讨论一些超级挑战。在这个版本中，我们报告Roberto Tauraso指出，在我们的文章末尾，我们所有的被证明的超同余都是已知的，除了一个，他提供了一个美丽的证明，可以在这里找到：arXiv：1606.05543。
英文摘要：
In a recent beautiful but technical article, William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences, namely those (like the Catalan and Motzkin sequences) that are expressible in terms of constant terms of powers of Laurent polynomials. We first give a leisurely exposition of their elementary but brilliant approach, and then extend it in two directions. The Laurent polynomials may be of several variables, and instead of single sums we have multiple sums. In fact we even combine these two generalizations! We conclude with some super-challenges. In this version we report that Roberto Tauraso pointed out that all our conjectured super-congruences, at the end of our article are already known, except one, for which he supplied a beautiful proof that can be found here: arXiv:1606.05543.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
