The Shi arrangements and the Bernoulli polynomials

Primitive derivations, Shi arrangements and Bernoulli polynomials

LetW be a finite irreducible real reflection group, which is a Coxeter group. A primitive derivation D, introduced and studied by K. Saito (e.g., [4]), plays a crucial role in the theory of differential forms with logarithmic poles along the Coxeter arrangement. For example, we may describe the contact order filtration of the logarithmic derivation module using the primitive derivations ([10, 1...

Bernoulli Basis and the Product of Several Bernoulli Polynomials

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...

Universal Bernoulli Polynomials And

Bernoulli polynomials