A New Approach to -Bernoulli Numbers and -Bernoulli Polynomials Related to -Bernstein Polynomials

A new approach to Bernoulli polynomials

Six approaches to the theory of Bernoulli polynomials are known; these are associated with the names of J. Bernoulli [2], L. Euler [4], E. Lucas [8], P. E. Appell [1], A. Hürwitz [6] and D. H. Lehmer [7]. In this note we deal with a new determinantal definition for Bernoulli polynomials recently proposed by F. Costabile [3]; in particular, we emphasize some consequent procedures for automatic c...

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...

On Bernoulli Sums and Bernstein Polynomials

In the paper we discuss a technology based on Bernstein polynomials of asymptotic analysis of a class of binomial sums that arise in information theory. Our method gives a quick derivation of required sums and can be generalized to multinomial distributions. As an example we derive a formula for the entropy of multinomial distributions. Our method simplifies previous work of Jacquet, Szpankowsk...

Some Identities of Bernoulli Numbers and Polynomials Associated with Bernstein 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