Bernoulli number
Introduction
The Bernoulli numbers are a sequence of rational numbers that play a significant role in number theory and analysis. They are named after Swiss mathematician Jacob Bernoulli, who made significant contributions to the field of calculus. The Bernoulli numbers appear in the Taylor series expansions of many functions, and they also appear in the Euler-Maclaurin formula, which provides a powerful method for approximating integrals and sums.
Definition
The Bernoulli numbers can be defined using the generating function:
- B(x) = \frac{x}{e^x - 1} = \sum_{n=0}^{\infty} B_n \frac{x^n}{n!}
where B_n denotes the nth Bernoulli number. The coefficients B_n in this series are the Bernoulli numbers. The generating function satisfies the functional equation B(x) = e^x B(1-x), which reflects the symmetry property B_n = (-1)^n B_{1-n} of the Bernoulli numbers.
Properties
The Bernoulli numbers have many interesting properties. For instance, they are alternating in sign, with B_1 = -1/2 and B_n = 0 for all odd n > 1. They also satisfy many interesting identities, such as the sum of the kth powers of the first n positive integers.
Recurrence Relation
The Bernoulli numbers satisfy the following recurrence relation:
- B_n = -\frac{1}{n+1} \sum_{k=0}^{n-1} \binom{n+1}{k} B_k
This recurrence relation provides a practical method for computing the Bernoulli numbers.
Sum of Powers
The Bernoulli numbers appear in the formula for the sum of the kth powers of the first n positive integers:
- \sum_{i=1}^{n} i^k = \frac{1}{k+1} \sum_{j=0}^{k} \binom{k+1}{j} B_j n^{k+1-j}
This formula was discovered by Jacob Bernoulli and generalized by Gauss and others.
Applications
The Bernoulli numbers have many applications in mathematics, particularly in number theory, analysis, and combinatorics.
Number Theory
In number theory, the Bernoulli numbers appear in the formulas for the sum of powers, in the study of Fermat's Last Theorem, and in the theory of modular forms.
Analysis
In analysis, the Bernoulli numbers appear in the Taylor series expansions of many functions, in the Euler-Maclaurin formula, and in the study of the Riemann zeta function.
Combinatorics
In combinatorics, the Bernoulli numbers count certain types of permutations and appear in the study of Stirling numbers.
History
The Bernoulli numbers were first introduced by Jacob Bernoulli in his book "Ars Conjectandi", published posthumously in 1713. However, they were studied earlier by Japanese mathematician Seki Takakazu, who discovered many of their properties.