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Bernoulli polynomials

In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions.

This article also discusses the Bernoulli polynomials and the related Euler polynomials, and the Bernoulli and Euler numbers.

Bernoulli polynomials

Representations

The Bernoulli polynomials B_n admit a variety of different representations. Which among them should be taken to be the definition may depend on one’s purposes.

Explicit formula

for n ≥ 0, where b_k are the Bernoulli numbers.

Generating functions

The generating function for the Bernoulli polynomials is

The generating function for the Euler polynomials is

Representation by a differential operator

The Bernoulli polynomials are also given by

where D = d/dx is differentiation with respect to x and the fraction is expanded as a formal power series. It follows that

cf. integrals below.

Representation by an integral operator

The Bernoulli polynomials are the unique polynomials determined by

 

The integral transform

 

on polynomials f, simply amounts to

This can be used to produce the inversion formulae below.

Explicit expressions for low degrees

The first few Bernoulli polynomials are:

 

欲探史實曾想讀歐拉大著，無奈語文不通方作罷︰

Institutiones calculi differentialis

Institutiones calculi differentialis (Foundations of differential calculus) is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential calculus. It consists of a single volume containing two internal books; there are 9 chapters in book I, and 18 in book II.

W. W. Rouse Ball (1888) writes that “this is the first textbook on the differential calculus which has any claim to be both complete and accurate, and it may be said that all modern treatises on the subject are based on it.”

Institutiones calculi differentialis

 

後聞早有英譯本，未讀不知不能說其事，乃今才曉此書網上有！