华蘅芳数在幂和问题中的新应用

Sum of Powers of Integers: An Application of Hua's Numbers

中文关键词:  自然效的幂的和  华氏数  组合模型  斯特灵数  华蘅芳.

英文关键词:formula of sum of powers of integers  Hua's numbers  combinatorial model  Stirling numbers  mathematician Hua Heng fang .

中文摘要:

      自然效的幂和问题具有悠久的历史,亦不乏现代的兴趣.一般学者不了解清代数学家华蘅芳的成果.本文改进了华氏数的定义;针对该问题建立了新的取盒-放球模型,给出幂和的组合解释;应用华氏数获得了简捷的幂和公式.文末介绍了华氏数的历史来源.

英文摘要:

      Hua Heng fang (1833-1902) was a famous mathematician in the end of Qing Dynasty . In his book Ji Jiao Shu (A Method of Finite Difference, 1870 '). Hua gave a formula of sum of powers of natural numbers using Hua's numbers. The study on sum of powers of natural numbers has a long history and a common interest today. Hua's numbers have good qualities but are not known by many math ematicians. Awaked by Hua's method only change one s ign in Hua's definition and get a new formula of sum of powers of integer in th is paper. This formula is very simple, and has some combinatorial sign ificance. A box taking and boll putting combinatorial model is established also.