A few remarks on Euler and Bernoulli

Discrete Dynamics in Nature and Society, 2011

We give some new identities on the Bernoulli and Euler numbers by using the bosonicp-adic integral onZpand reflection symmetric properties of Bernoulli and Euler polynomials.

Bernoullian treatise, 2023

Starting from a few elements, binomial matrices and vectors, without resorting to the traditional expansion in power series, polynomials and Bernoulli numbers are defined, proving various theorems on them. Along the way, the classic problem of the sum of powers of successive integers, which is generalized to sums of powers having any arithmetic progression as a basis, is also solved, in various ways,. The itinerary develops through numerous propositions rigorously linked on the classic model of Euclid's Elements. Numerous examples and hypertext references have the didactic purpose of making reading easier.

2014

We investigate generalized binomial expansions that arise from two-dimensional sequences satisfying a broad generalization of the triangular recurrence for binomial coefficients. In particular, we present a new combinatorial formula for such sequences in terms of a ‘shift by rank’ quasi-expansion based on ordered set partitions. As an application, we give a new proof of Dilcher’s formula for expressing generalized Bernoulli polynomials in terms of classical Bernoulli polynomials.

J. Integer Seq., 2018

In this note we prove combinatorially some new formulas connecting poly-Bernoulli numbers with negative indices to Eulerian numbers.

The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the Srivastava-Pintér addition theorem is obtained. We give new identities involving q-Bernstein polynomials.

Linear Algebra and its Applications, 2014

In this paper we establish plenty of number theoretic and combinatoric identities involving generalized Bernoulli and Stirling numbers of both kinds. These formulas are deduced from Pascal type matrix representations of Bernoulli and Stirling numbers. For this we define and factorize a modified Pascal matrix corresponding to Bernoulli and Stirling cases.

An overlooked formula of E. Lucas for the generalized Bernoulli numbers is proved using generating functions. This is then used to provide a new proof and a new form of a sum involving classical Bernoulli numbers studied by K. Dilcher. The value of this sum is then given in terms of the Meixner-Pollaczek polynomials.

Tamkang Journal of Mathematics

Here an attempt has been made to extend the Bernoulli, Euler and Eulerian polynomials in multiplication theorem and finite difference formula have been established.

arXiv: Combinatorics, 2015

A sequence inverse relationship can be defined by a pair of infinite inverse matrices. If the pair of matrices are the same, they define a dual relationship. Here presented is a unified approach to construct dual relationships via pseudo-involution of Riordan arrays. Then we give four dual relationships for Bernoulli numbers and Euler numbers, from which the corresponding dual sequences of Bernoulli polynomials and Euler polynomials are constructed. Some applications in the construction of identities of Bernoulli numbers and polynomials and Euler numbers and polynomials are discussed based on the dual relationships.

2010

We introduce elliptic analogues to the Bernoulli ( resp. Euler) numbers and functions. The first aim of this paper is to state and prove that our elliptic Bernoulli and Euler functions satisfied Raabe’s formulas (cf. Theorems 3.1.1, 3.2.1). We define two kinds of elliptic Dedekind-Rademacher sums, in terms of values of our elliptic Bernoulli (resp. Euler) functions. The second aim of this paper is to state and prove a Dedekind reciprocity Law for our elliptic Dedekind-Rademacher sums (cf.Theorems 4.3.1, 4.4.1). Our results recover the well known classical results.