高阶Bernoulli多项式

THE HIGHER ORDER BERNOULLI’S POLYNOMIAL

本文从事高阶Bernoulli数、高阶Bernoulli多项式及其性质的研究,揭示n阶ν次Bernoulli数为一卷积Bν（n）=Bν（n-k）*Bν（k） （k<n）另经论证下标算子1表示的n阶ν次Bernoulli多项式的显式为Bν（n）（X）=（■+X）（ν）Bν（n） ν=1,2,…并在此基础上论证Bν（n）（X）的差分、微分、积分、…等性质,发现Bν（n）（X）与Bν（X）同构.

In this paper,the heigher-order Bernoulli's numbers,its polynomial and character are studied.Their results have been obtained as following: < 1> N-order v-degree Bernoulli's numbers may be expressed as B_v^((n)) = B_v^((n-k)) *B_v^((k)),where k<n; < 2 > N-order v-degree Bernoulli's polynomials with a subscript operater '1' could be expressd B_v^((n)) (x) = (1+x)~vB_v^((n)),where v =1,2,…. Futhermore,many characteristics of B_v^((n))(x),such as its defference, differential and integral have also been obtained.Finally,it is showed in this paper that B_v^((n)) (x ) and B_v ( x ) are isomorphic.