Generalised Bernoulli polynomials and series

Old and New Identities for Bernoulli Polynomials via Fourier Series

The Bernoulli polynomials Bk restricted to 0, 1 and extended by periodicity have nth sine and cosine Fourier coefficients of the formCk/n . In general, the Fourier coefficients of any polynomial restricted to 0, 1 are linear combinations of terms of the form 1/n . If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a...

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

Congruences involving Bernoulli polynomials

Let {Bn(x)} be the Bernoulli polynomials. In the paper we establish some congruences for Bj(x) (mod p n), where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p-regular functions, the congruences for h(−sp) (mod p) (s = 3, 5, 8, 12) and the sum P k≡r (mod m) p k , where h(d) is the class number of the quadratic field Q(d) of discriminant d...