Related papers: Lecture Notes, Bernoulli Polynomials and Applicati…

This is an expository survey on the theory of Bernstein-Sato polynomials with special emphasis in its recent developments and its importance in commutative algebra.

In this paper we use computational method based on operational point of view to prove a new generating function of exponential polynomials. We give its applications involving geometric polynomials, Bernoulli and Euler numbers.

In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, Dere and Simsek have studied umbral calculus related…

The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of…

We introduce and study a `level two' generalization of the poly-Bernoulli numbers, which may also be regarded as a generalization of the cosecant numbers. We prove a recurrence relation, two exact formulas, and a duality relation for…

We give direct and inverse theorems for the weighted approximation of functions with inner singularities by combinations of Bernstein polynomials.

We define the toric Newton spectrum of a polynomial and we give some applications in singularity theory, combinatorics and mirror symmetry.

We generalise the Bernoulli numbers to include the case where the index may be a continuous variable.

We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain infinite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums.…

In this paper we consider the notions of binomial thinning, binomial mixing, their generalizations, certain interplay between them, associated limit theorems and provide various examples.

This note gives an elementary exposition of a variant of the spread polynomials in terms of Fibonacci and Lucas polynomials.

In this work, a new approach has been developed to obtain numerical solution of linear Volterra type integral equations by obtaining asymptotic approximation to solutions. Using the classical Bernoulli polynomials, a set of orthonormal…

In the first part of this expository paper, we present and discuss the interplay of Dirichlet polynomials in some classical problems of number theory, notably the Lindel\"of Hypothesis. We review some typical properties of their means and…

The main purpose and motivation of this article is to create a linear transformation on the polynomial ring of rational numbers. A matrix representation of this linear transformation based on standard fundamentals will be given. For some…

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…

We show that certain weighted Fibonacci and Lucas series can always be expressed as linear combinations of polylogarithms. In some special cases we evaluate the series in terms of Bernoulli polynomials, making use of the connection between…

We consider the Bernoulli polynomials of the second kind, which can be related to the generalised Bernoulli polynomials $B_n^{(n)}(z)$. The asymptotic expansions of the scaled polynomials $B_n^{(n)}(nz)$ are obtained as $n\to\infty$ when…

This paper shows that a finite discrete convolution involving Stirling numbers of both kinds and harmonic numbers can be expressed in terms of the Bernoulli numbers. As applications of this expression, the linear recurrence relation for the…

We construct the new q-extension of Bernoulli numbers and polynomials in this paper. Finally we consider the q-zeta functions which interpolate the new q-extension of Bernoulli numbers and polynomials.

In the present paper, we introduce Eulerian polynomials with a and b parameters and give the definition of them. By using the definition of generating function for our polynomials, we derive some new identities in Theory of Analytic…