Treatise on the Cálculus of Finite Differences

2004

A relation showing that the Grünwald-Letnikov and generalized Cauchy derivatives are equal is deduced confirming the validity of a well known conjecture. Integral representations for both direct and reverse fractional differences are presented. From these the fractional derivative is readily obtained generalizing the Cauchy integral formula.

Publications de l'Institut Mathematique, 2015

We obtain a new recurrence formula for sequences of divided differences. In a particular case, the recurrence formula simplifies the classical Newton-Girard identities relating power sums and elementary symmetric polynomials.

Archive for history of exact sciences, 1989

The Mathematical Gazette, 2002

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

1989

Functions can often be de ned by recursive equations such as these:

1987

This document introduces a notion of functional differences in which the difference of a function f with respect to a function h is that function g that describes how the value of f changes when its argument is altered by h: f (h (x)= g (f (x)). The author introduces the inverse operation of functional integration and derive useful properties of both operations. The result is a calculus that facilitates derivation and reasoning about recursive programs. This is illustrated in a number of simple examples.

Applicable Analysis and Discrete Mathematics , 2020

In this study, depending on the upper and the lower indices of the hyperhar-monic number h (r) n , nonlinear recurrence relations are obtained. It is shown that generalized harmonic numbers and hyperharmonic numbers can be obtained from derivatives of the binomial coefficients. Taking into account of difference and derivative operators, several identities of the harmonic and hy-perharmonic numbers are given. Negative-ordered hyperharmonic numbers are defined and their alternative representations are given.

Formalized Mathematics, 2008

In this article, we give some important properties of forward difference, backward difference, central difference and difference quotient and forward difference, backward difference, central difference and difference quotient formulas of some special functions .