Related papers: Lecture Notes, Bernoulli Polynomials and Applicati…
This work is a study of polynomial compositions having a fixed number of terms. We outline a recursive method to describe these characterizations, give some particular results and discuss the general case. In the final sections, some…
In the present paper, we obtain new interesting relations and identities of the Apostol-Bernoulli polynomials of higher order, which are derived using a Bernoulli polynomial basis. Finally, by utilizing our method, we also derive formulas…
Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we develop general theorems on permutation polynomials over finite fields. As a…
We review and discuss some results on the representation of Bernoulli, poly-Bernoulli numbers, and Bernoulli and Cauchy polynomials in terms of Stirling numbers of the first or second kind, or in terms of r-Stirling numbers.
This article is a survey of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of Analysis. Some new properties are included and several Analysis-related applications are mentioned.
We investigate paths in Bernoulli's triangles, and derive several relations linking the partial sums of binomial coefficients to the Fibonacci numbers.
The main aim of this article is a careful investigation of the asymptotic behavior of zeros of Bernoulli polynomials of the second kind. It is shown that the zeros are all real and simple. The asymptotic expansions for the small, large, and…
We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using…
In this paper, we consider the degenerate Carlitz q-Bernoulli numbers and polynomials and we investigate some properties of those polynomials.
A new method of algebraic nature is proposed for the study of the asymptotic properties of special polynomials. The technique we foresee is based on the use of umbral operators, allowing a unified treatment of a large body of polynomial…
In this paper, we present a probabilistic extension of the Fubini polynomials and numbers associated with a random variable satisfying some appropriate moment conditions. We obtain the exponential generating function and an integral…
In this paper, we consider the new q-extension of Frobenius-Euler numbers and polynomials and we derive some interesting identities from the orthogonality type properties for the new q-extension of Frobenius-Euler polynomials. Finally we…
We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials.
Recently, Komastu introduced the concept of poly-Cauchy numbers and polynomials which generalize Cauchy numbers and polynomials. In this paper, we introduce new generaliza- tion of poly-Cauchy and poly-Bernoulli numbers and polynomials.…
Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication…
The aim of this note is to provide a simple proof of some well-known identities and recurrences relating classical Bernoulli and Euler numbers by using the Abel sum of the divergent series $\sum_{n=0}^\infty (-1)^{n} (n+1)^k$, $k$ a…
We introduce generalized hypergeometric Bernoulli numbers for Dirichlet characters. We study their properties, including relations, expressions and determinants. At the end in Appendix we derive first few expressions of these numbers.
We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method…
In this paper, we study the binomial sum $S_{n}(q):=% \overset{n}{\underset{k=0}{\sum }}a_{k}\binom{n}{k}\left( 1-q\right) ^{k}q^{n-k}$ for a given sequence $\left( a_{n}\right) $ of real or complex numbers. We express $S_{n}(q)$ in…
In the present paper we generalize the Eulerian numbers (also of the second and third orders). The generalization is connected with an autonomous first-order differential equation, solutions of which are used to obtain integral…