Distribution of the primes involving the ceiling function

The distribution of the primes of the forms [Formula: see text] and [Formula: see text] are studied extensively, where [Formula: see text] denotes the largest integer not exceeding [Formula: see text]. In this paper, we will consider several new type problems on the distribution of the primes involving the ceiling (floor) function. For any real number [Formula: see text] with [Formula: see text], let [Formula: see text] be the number of integers [Formula: see text] with [Formula: see text] such that [Formula: see text] is prime and let [Formula: see text] be the number of primes [Formula: see text] for which there exists an integer [Formula: see text] with [Formula: see text] such that [Formula: see text], where [Formula: see text] denotes the least integer not less than [Formula: see text]. These are closely related to the number of the prime factors of the denominator of the Bernoulli polynomial [Formula: see text]. In this paper, we study asymptotic properties of [Formula: see text] and [Formula: see text]. The methods in this paper are also effective for corresponding distribution functions of the primes involving the floor function.

Truncated euler polynomials

We define a truncated Euler polynomial Em,n(x) as a generalization of the classical Euler polynomial En(x). In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial.

Bernoulli polynomials collocation for weakly singular Volterra integro-differential equations of fractional order

This paper is concerned with a numerical procedure for fractional Volterra integro-differential equations with weakly singular kernels. The fractional derivative is in the Caputo sense. In this study, Bernoulli polynomial of first kind is used and its matrix form is given. Then, the matrix form based on the collocation points is constructed for each term of the problem. Hence, the proposed scheme simplifies the problem to a system of algebraic equations. Error analysis is also investigated. Numerical examples are announced to demonstrate the validity of the method.

Proof of some congruences conjectured by Z.-W. Sun

In this paper, we show that for any prime [Formula: see text], [Formula: see text] and [Formula: see text] where [Formula: see text] denotes the Bernoulli polynomial of degree [Formula: see text]. And we prove that [Formula: see text] and [Formula: see text] where [Formula: see text] stands for the [Formula: see text]th Bernoulli number. This confirms several conjectures of Z.-W. Sun.

RADEMACHER CHAOSES AND BERNOULLI POLYNOMIALS

In this paper we prove that any Bernoulli polynomial of even (odd) order is an absolutely convergent series of functions from some Rademacher chaoses, each of them is of even (odd) order

Bernoulli and poly-Bernoulli polynomial convolutions and identities of p-adic Arakawa–Kaneko zeta functions

We evaluate the ordinary convolution of Bernoulli polynomials in closed form in terms of poly-Bernoulli polynomials. As applications we derive identities for [Formula: see text]-adic Arakawa–Kaneko zeta functions, including a [Formula: see text]-adic analogue of Ohno’s sum formula. These [Formula: see text]-adic identities serve to illustrate the relationships between real periods and their [Formula: see text]-adic analogues.