Bernoulli F-polynomials and Fibo–Bernoulli matrices

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...

Determinants and permanents of Hessenberg matrices and generalized Lucas polynomials

In this paper, we give some determinantal and permanental representations of generalized Lucas polynomials, which are a general form of generalized bivariate Lucas p-polynomials, ordinary Lucas and Perrin sequences etc., by using various Hessenberg matrices. In addition, we show that determinant and permanent of these Hessenberg matrices can be obtained by using combinations. Then we show, the ...