Let p be a fixed odd prime. Throughout this paper, Zp,Qp,Cp will, respectively, denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp. The p-adic absolute value | |p on Cp is normalized so that |p|p 1/p. Let Z>0 be the set of natural numbers and Z≥0 Z>0 ∪ {0}. As is well known, the Bernoulli polynomials Bn x are defined by the ge...

The current article focus on the ordinary Bernoulli, Euler and Genocchi numbers and polynomials. It introduces a new approach to obtain identities involving these special polynomials and numbers via generating functions. As an application of the new approach, an easy proof for the main result in [6] is given. Relationships between the Genocchi and the Bernoulli polynomials and numbers are obtai...

A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number theoretical properties. A class of Euler-type polynomials is also presented. © 2007 Elsevier Inc. All rights reserved.

We investigate Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials using the Lipschitz summation formula and obtain their integral representations. We give some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz zeta function. We also derive the integral representations for the classical Bernoulli and Euler polynomials and related known ...

The main object of this paper is to investigate the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials. We first establish two relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials. It can be found that many results obtained before are special cases of these two relationships. Moreover, we have a study on the sums of products of the Apostol-Bernoulli...

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials Bn(x;λ) in detail. The starting point is their Fourier series on [0, 1] which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain ...

Energy harvesting by using piezoelectric materials is one of the most widely used techniques for conversion waste energy into useful. Using this technique, generated vibration from machines can be converted useful electrical energy. In paper, an system that supplies power low-power consumption devices has been designed. The experimental model consists a rotating machine generates mechanical vib...

In this paper, we use Hermite cubic finite elements to approximate the solutions of a nonlinear Euler– Bernoulli beam equation. The equation is derived from Hollomon’s generalized Hooke’s law for work hardening materials with the assumptions of the Euler–Bernoulli beam theory. The Ritz–Galerkin finite element procedure is used to form a finite dimensional nonlinear program problem, and a nonlin...

Let B n = b 1 +. .. + b n , n ≥ 1 where b 1 , b 2 ,. .. are independent Bernoulli random variables. In relation with the divisor problem, we evaluate the almost sure asymptotic order of the sums N n=1 d θ,D (B n), where d θ,D (B n) = #{d ∈ D, d ≤ n θ : d|B n } and D is a sequence of positive integers.

Let X1, X2, X3, . . . be i.i.d. random variables (Here ”i.i.d.” means ”independent and identically distributed” ), s.t. Pr[Xi = 1] = p, Pr[Xi = 0] = 1 − p. Xi is also called Bernoulli random variable. Let Sn = X1 + · · · + Xn. We will be interested in the random variable Sn which is called Binomial random variable (Sn ∼ B(n, p)). If you toss a coin for n times, and Xi = 1 represents the event t...