Several formulas for Bernoulli numbers and polynomials

Explicit Formulas for Bernoulli and Euler Numbers

Explicit and recursive formulas for Bernoulli and Euler numbers are derived from the Faá di Bruno formula for the higher derivatives of a composite function. Along the way we prove a result about composite generating functions which can be systematically used to derive such identities.

Congruences concerning Bernoulli numbers and Bernoulli polynomials

Let {Bn(x)} denote Bernoulli polynomials. In this paper we generalize Kummer’s congruences by determining Bk(p−1)+b(x)=(k(p − 1) + b) (modp), where p is an odd prime, x is a p-integral rational number and p − 1 b. As applications we obtain explicit formulae for ∑p−1 x=1 (1=x ) (modp ); ∑(p−1)=2 x=1 (1=x ) (modp ); (p − 1)! (modp ) and Ar(m;p) (modp), where k ∈ {1; 2; : : : ; p− 1} and Ar(m;p) i...

Bernoulli Basis and the Product of Several Bernoulli Polynomials

Identities of Bernoulli Numbers and Polynomials

In the field of Laurent series Q((T)), the series B = T/(e T − 1) is contained in the formal power series ring Q[[T ]]. The i-th Bernoulli number B

Generalizations of Bernoulli Numbers and Polynomials

The concepts of Bernoulli numbers B n , Bernoulli polynomials B n (x), and the generalized Bernoulli numbers B n (a, b) are generalized to the one B n (x; a, b, c) which is called the generalized Bernoulli polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between B n , B n (x), B n (a, b), and B n (x; a, b, c) are established.