Bernoulli Polynomials – Science Solidaire

1. Introduction

Bernoulli polynomials, introduced by Swiss mathematician Jacob Bernoulli in the 17^th century, form a fundamental family of polynomials in mathematics. They appear in various fields such as number theory, mathematical analysis, and symbolic computation.

Defined through a generating function, these polynomials are closely related to Bernoulli numbers, which are coefficients appearing in Taylor series expansions of certain trigonometric functions. Their importance is also evident in power sum formulas and their relationship with the Riemann zeta function.

The aim of this article is to explore the fundamental properties of Bernoulli polynomials, their relationships with other mathematical objects, and their modern applications.

2. Definition and Properties

Bernoulli polynomials $B_n(x)$ are defined by the following generating function:

$\frac{t e^{xt}}{e^t - 1} = \sum_{n=0}^{\infty} B_n(x) \frac{t^n}{n!}, \quad \text{for } |t| < 2\pi.$

The first Bernoulli polynomials are:

$B_0(x) = 1, \quad B_1(x) = x - \frac{1}{2}, \quad B_2(x) = x^2 - x + \frac{1}{6}.$

These polynomials satisfy a fundamental recurrence relation:

$B_n'(x) = n B_{n-1}(x).$

They are also related to Bernoulli numbers $B_n$ , defined by $B_n(0) = B_n$ .

3. Applications in Mathematics

Bernoulli polynomials play a crucial role in various areas of mathematics. Here are some notable applications:

Euler-Maclaurin summation formula: This formula connects discrete sums to integrals and involves Bernoulli polynomials in the correction terms. It is particularly useful for series approximation and sum calculations.
Power sum calculations: Bernoulli polynomials allow expressing sums of p-th powers of the first n natural numbers. For example, the sum of squares can be calculated using appropriate values of Bernoulli polynomials.
Number theory: They appear in the study of the Riemann zeta function and are used to analyze prime number distribution. Special values of the zeta function at negative integers are related to Bernoulli numbers.

4. Examples and Illustrations

To illustrate the use of Bernoulli polynomials, consider calculating the sum of fourth powers of the first $n$ natural numbers:

$S = \sum_{k=1}^n k^4$

Using Bernoulli polynomials, this sum can be expressed as:

$S = \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}$

This formula allows efficient calculation of the power sum without adding each term individually.

1. Introduction

2. Definition and Properties

3. Applications in Mathematics

4. Examples and Illustrations

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