The Haruki-Rassias and related integral representations of the Bernoulli and Euler polynomials

Haruki and Rassias [H. Haruki, T.M. Rassias, New integral representations for Bernoulli and Euler polynomials, J. Math. Anal. Appl. 175 (1993) 81-90] found the integral representations of the classical Bernoulli and Euler polynomials and proved them by making use of the properties of certain functional equation. In this sequel, we rederive, in a completely different way, the results of Haruki and Rassias and deduce related and new integral representations. Our proofs are quite simple and remarkably elementary.     

On hypergeometric Bernoulli numbers and polynomials

We provide several properties of hypergeometric Bernoulli numbers and polynomials, including sums of products identity, differential equations and recurrence formulas.     

On the decomposability of linear combinations of Bernoulli polynomials

Monatshefte für Mathematik - In the present paper we describe the complete decomposition (over $$\mathbb {C}$$ ) of linear combinations of the form $$\begin{aligned} R_n(x)=B_n(x)+cB_{n-2}(x)...     

Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.

     

On the real zeroes of the Hurwitz zeta-function and Bernoulli polynomials

The behaviour of real zeroes of the Hurwitz zeta-function

     

Generalization of Bernoulli polynomials

The Bernoulli polynomials are generalized and some properties of the resulting generalizations are presented.     

On Appell sequences of polynomials of Bernoulli and Euler type

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.     

Some properties on the integral of the product of several Euler polynomials

Abstract

In this paper, we study the formula for a product of two Euler poly-nomials. From this study, we derive some formulae for the integral of the product of two or more Euler polynomials.     

On conjugate product of complex polynomials

Some concepts, such as divisibility, coprimeness, in the framework of ordinary polynomial product are extended to the framework of conjugate product. Euclidean algorithm for obtaining greatest common divisors in the framework of conjugate product is also established. Some criteria for coprimeness are established.     

Identities for degenerate Bernoulli polynomials and Korobov polynomials of the first kind

In this paper, we derive five basic identities for Sheffer polynomials by using generalized Pascal functional and Wronskian matrices. Then we apply twelve basic identities for Sheffer polynomials, seven from previous results, to degenerate Bernoulli polynomials and Korobov polynomials of the first kind and get some new identities. In addition, letting λ→ 0 in such identities gives us those for Bernoulli polynomials and Bernoulli polynomials of the second kind.     

Higher-order convolutions for Bernoulli and Euler polynomials

We prove convolution identities of arbitrary orders for Bernoulli and Euler polynomials, i.e., sums of products of a fixed but arbitrary number of these polynomials. They differ from the more usual convolutions found in the literature by not having multinomial coefficients as factors. This generalizes a special type of convolution identity for Bernoulli numbers which was first discovered by Yu. Matiyasevich.     

New identities involving Bernoulli and Euler polynomials

Using the finite difference calculus and differentiation, we obtain several new identities for Bernoulli and Euler polynomials; some extend Miki's and Matiyasevich's identities, while others generalize a symmetric relation observed by Woodcock and some results due to Sun.     

On Galois spectra of polynomials with integral parameters

We prove that there exists a polynomial F(x, t) with rational coefficients, whose degree with respect to x is equal to 4, such that for every integer a, the Galois group of the decomposition field of the polynomial F(x, a) is not the dihedral group, but any other transitive subgroup of the group S₄ can be represented as the Galois group of the decomposition field of the polynomial F(x, a) for a certain integer a. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 275–280.