Euler-Maclaurin Formula

Notes on Number Theory and Discrete Mathematics, 2021

In this paper, we give several explicit formulas involving the n-th Euler polynomial E n (x). For any fixed integer m ≥ n, the obtained formulas follow by proving that E n (x) can be written as a linear combination of the polynomials x n , (x + r) n ,. .. , (x + rm) n , with r ∈ 1, −1, 1 2. As consequence, some explicit formulas for Bernoulli numbers may be deduced.

Historia Mathematica, 1998

Wallis's method of interpolation attracted the attention of the young Euler, who obtained some important results. The problem of interpolation led Euler to formulate the problem of integration, i.e., to express the general term of a series by means of an integral. The latter problem was connected to the question of expressing the sum of a series using an integral. The outcome of this research was Euler's derivation of what would later become known as the Euler–Maclaurin formula. Euler subsequently returned to interpolation and formulated the theory of inexplicable functions including the gamma function.The methods used by Euler illustrate well the principles of 18th-century analysis. Eulerian procedures are based upon the notion of geometric quantity. A function is actually conceived as the expression of a quantity and, for this reason, it intrinsically possesses properties we can term continuity, differentiability, Taylor expansion. These correspond to the usual properties of a curve which has “regular” characteristics (lack of jumps, presence of tangents, curvature radius, etc.). They have a “figural” clarity. Although Eulerian analysis remains rooted in geometry, it dispenses with figural representation: it is substantially nonfigural geometry. Reasoning with figures (which integrates the proof in classical geometry) is replaced by reasoning with analytic symbols. These are general because they do not represent a particular quantity and are not subjected to restrictions, but are an abstract representation of quantity.Copyright 1998 Academic Press.Il problema dell'interpolazione wallisiana attrasse l'attenzione del giovane Euler. Egli ottenne rapidamente risultati di grande interesse e proseguı̀ le sue ricerche formulando il problema dell'integrazione consistente nell'esprimere il termine generale di una serie mediante un integrale. Quest'ultimo problema ben presto si evolse nella questione di esprimere la somma di una serie mediante un integrale che guidò Euler nella derivazione della formula sommatoria oggi detta di Euler-Maclaurin. Euler ritornò in seguito sul problema dell'interpolazione sviluppando la teoria delle funzioni inesplicabili che comprendono come caso particolare la gamma.I metodi usati da Euler sono di grande interesse per una comprensione dell'analisi settecen- tesca. Infatti alla base delle procedure euleriane vi è la nozione di quantità geometrica, la quale comporta che la funzione, intesa come espressione della quantità, abbia intrinsecamente proprietà che, con linguaggio moderno, possiamo chiamare continuità, differenziabilità, sviluppabilità in serie de Taylor, e che corrispondono alle proprietà tipiche di una curva dotata di “regolarità” (non fare salti, esistenza della tangente, del raggio di curvatura, ecc.). Queste proprietà hanno un'immediata evidenza nelle curve comunemente studiate, un'evidenza, per cosi dire, figurale. L'analisi euleriana conserva un forte contenuto geometrico, ma elimina la rappresentazione figurale: sostanzialmente essa appare come una geometria non figurale. Il ragionamento sulla figura (che nella geometria classica integrava la dimostrazione) viene ora sostitutito dal ragionamento sui simboli analitici, i quali sono generali, perché non rappresentano questa o quella particolare quantità e non sone soggetti a limitazioni di sorta, ma sono una rappresentazione astratta della quantità.Copyright 1998 Academic Press.Le problème d'interpolation de Wallis attira l'attention du jeune Euler, qui obtint rapidement des résultats de grand intérêt. Il amena Euler à formuler le problème d'intégration consistant à exprimer le terme général d'une série par une intégrale. Ce dernier problème se transforma rapidement en celui d'exprimer la somme d'une série par une intégrale, menant à la formule appelée de nos jours la formule d'Euler-Mac Laurin. Euler reprit ultérieurement le problème d'interpolation, formulant la théorie des fonctions “inexplicables” qui incluent en particulier la fonction Gamma.Les méthodes utilisées par Euler ont un grand intérêt pour comprendre l'analyse du XVIIIe siècle. En effet, à la base des procédures eulériennes, on trouve la notion de quantité géométrique, qui implique que la fonction, comprise comme une expression de cette quantité, possède intrinsèquement certaines propriétés—en langage moderne la continuité, la différentiabilité, le développement en séries de Taylor—correspondant aux propriétés typiques d'une courbe dotée de régularité—ne pas faire de sauts, avoir une tangente, un rayon de courbure. Ces propriétés étaient évidentes pour les courbes communément étudiées, une évidence pour ainsi dire figurale. L'analyse eulérienne conserve un important contenu géométrique, mais élimine la représentation figurale: concrêtement, elle apparaı̂t comme une géométrie non figurale. Le raisonnement sur la figure, qui, dans la géométrie classique, intégrait la démonstration, est ici remplacé par le raisonnement sur les symboles analytiques: ceux-ci sont généraux parce qu'ils ne représentent pas une quantité particulière et ne sont pas soumis à des restrictions, mais sont une représentation abstraite de la quantité.Copyright 1998 Academic Press.AMS 1991 subject classifications: 01A50

Math. Comp, 1967

Some elementary methods are described which may be used to calculate tangent numbers, Euler numbers, and Bernoulli numbers much more easily and rapidly on electronic computers than the traditional recurrence relations which have been used for over a century. These methods have been used to prepare an accompanying table which extends the existing tables of these numbers. Some theorems about the periodicity of the tangent numbers, which were suggested by the tables, are also proved.

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.

Journal of Approximation Theory, 2011

Hurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning Bernoulli and Euler polynomials.

In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications to these polynomials are presented, including a unified approach to the asymptotic expansion of the error term in many numerical quadrature formulae, and many new and sharp inequalities, that bound some trigonometric sums.

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.

Indian Journal of Pure and Applied Mathematics, 2020

By examining higher derivatives of hyperbolic functions, we derive monomial and binomial representation formulae, that are utilized to establish several multiple convolution identities for the Bernoulli numbers B n , Euler numbers E n and two variants of B n

2013

In the paper, by establishing a new and explicit formula for computing the n-th derivative of the reciprocal of the logarithmic function, the author presents new and explicit formulas for calculating Bernoulli numbers of the second kind and Stirling numbers of the first kind. As consequences of these formulas, a recursion for Stirling numbers of the first kind and a new representation of the reciprocal of the factorial n! are derived. Finally, the author finds several identities and integral representations relating to Stirling numbers of the first kind.

2016

In this lecture notes we try to familiarize the audience with the theory of Bernoulli polynomials; we study their properties, and we give, with proofs and references, some of the most relevant results related to them. Several applications to these polynomials are presented, including a unified approach to the asymptotic expansion of the error term in many numerical quadrature formulae, and many new and sharp inequalities, that bound some trigonometric sums.