Bernoulli Numbers And Continuous Integer Powers

We define sums of powers Sm(n) by the formula: (m, n posi-tive integers). we know that S2m-1(n) is a polynomial function of S1(n) and its coefficients have a relation to the Bernoulli numbers.S2m(n) is the product of the polynomial function of S1(n) and S2 (n) and its coefficients have a relation to the Bernoulli numbers. In this thesis, we give the new proof.Firstly, we find the polynomial satisfy Hence the polynomial Pm(n) can be expressed by the polynomial of Si(n) and the polynomial Qm(n) can be expressed by the product of the polynomial of Si (n) and S2 (n). Because of the formula and we prove the above conclusion.Secondly, we introduce some equivalent definitions of Stirling numbers and give a new proof of higher derivation formula of composite functions. Using this formula and Bell polynomials, we prove the relations of Bernoulli numbers and Stirling numbers.