明安图与Catalan数

Ming Antu and Catalan Numbers

英文关键词:Ming Antu  Catalan numbers  combinatorics  counting function.

中文摘要:

      中国数学家明安图在其《割圜密率捷法》中最先应用了Catalan数,取得优秀的研究成果.本文简介明安图的计数成就和Catalan数,综述国内外对明安图应用该数的研究.特别地,近两年来英国的Larcombe发表了5篇文章,对明安图的成果——包含Catalan数的sin（2pa）展开式,加以推广,并给出明安图确定Catalan 数的第二种方法的严格代数证明.

英文摘要:

      Ming Antu, a famous mathematician in Qing Dynasty, had obtained important achievements in counting theory. Ming Antu and Catalan Numbers are described concisely in this article. The status of study on Ming Antu and his excellent achievement-Catalan Numbers both in and outside China is showed. Recently, P. J. Larcombe has made further studies on Ming Antu's counting achievement, and drawn a conclusion of expanding the function sin (2pa) on the base of Ming's study. The precise combination of Catalan Numbers is involved in this expansion. Moreover, he gave a strict algebraic proof to the second result of Ming on Catalan Numbers. These historical problems become a new starting point for current research.