Natural Numbers Power And The Equivalent Of Three Typical Formulas And The Analyze Of Algorithms Complexity

Bernoulli numbers, Stirling numbers and Euler numbers have a wide range of applications in many fields such as combinatorics, function theory, theoretical physics, approximate calculation, and so on. And the summation of powers of integers, as an old topic, has attracted many scholars. In digital images, Euler numbers can be used to describe the structure of the objects remaining characteristics of them unchanged. In discrete mathematics, these special numbers have the combinatorial meanings. They can also be used to count in meteorology, combinatorial optimization, random graph, Ramsey theory, etc., Donald E. Knuth, renowned computer scientist, from Stanford University, in his famous book The Art of Computer Programming, specifically designed programs for calculating Euler numbers and Bernoulli numbers. Research about power sum has experienced more than 2000 years. Famous Chinese mathematicians, Chen Jingrun, Li Jianyu and others, made a great number of researcher, which results in many advanced achievements far ahead. The power summation of natural numbers can be expressed respectively by Euler numbers, Bernoulli numbers, and Stirling numbers. But these three expressions have not been proved to be identical to each other. Moreover, the analysis for the algorithm complexity also has not been given yet so far. Mr. Xu Lizhi, famous Chinese mathematician, suggests that solving the two problems could be a thesis for master degree in his letter to Professor Luo Jianjin from Inner Mongolia Normal University. This implies that this kind of work is so important and the central work of my thesis is about it.This thesis discusses the origin and development of power summation, introduces the main achievements in the last 2000 years about it, based on which, the author recalls the researches related to the two kinds of Stirling numbers, Euler numbers, and Bernoulli numbers, proves some main results about the identity between the expressions of power summation with Euler numbers, Stirling numbers, and Bernoulli numbers, and finally gives the discussion about the complexity of the algorithm.