Bernoulli number

In mathematics, the Bernoulli numbers are a special sequence of rational numbers that show up in many different areas. They are important in studying things like the Taylor series of the tangent and hyperbolic tangent functions, formulas for adding up powers of numbers, and even in understanding the Riemann zeta function.

The first Bernoulli numbers have specific values, and they follow a pattern: for every odd number greater than 1, the Bernoulli number is zero. These numbers are connected to Bernoulli polynomials and have interesting properties that make them useful in many mathematical problems.

Bernoulli numbers were discovered around the same time by two great thinkers: Swiss mathematician Jacob Bernoulli and Japanese mathematician Seki Takakazu. Even famous computer pioneer Ada Lovelace wrote about how to generate these numbers using an early computer idea, making Bernoulli numbers part of one of the first computer programs ever described.

Notation

The symbol ± in this article helps show two different ways to write Bernoulli numbers. Only the term where n = 1 is affected:

• B⁻_n with B⁻₁ = −⁠1/2⁠ (OEIS: A027641 / OEIS: A027642) is the way recommended by NIST and many modern books.
• B⁺_n with B⁺₁ = +⁠1/2⁠ (OEIS: A164555 / OEIS: A027642) was used in older writings and, since 2022, by Donald Knuth.

You can change from one way to the other using a simple rule. For numbers where n is 2 or more, you can usually ignore the difference. Also, for all odd numbers greater than 1, B_n equals zero, so many formulas only use the even-numbered Bernoulli numbers.

History

Bernoulli numbers are a sequence of rational numbers that appear in many areas of mathematics. They were first introduced by Jacob Bernoulli in 1713 in his book Ars Conjectandi. These numbers help in calculating sums of powers of integers and appear in various mathematical formulas.

The concept originated from the need to find quick ways to compute sums of powers, like adding up the squares or cubes of numbers. Early mathematicians such as Pythagoras, Archimedes, and others studied these sums, but it was Jacob Bernoulli who found a unified approach using what we now call Bernoulli numbers. His work provided a formula that applies to all such sums, making calculations much easier.

Definitions

Bernoulli numbers are a special set of numbers in mathematics that show up in many different areas. There are several ways to describe them, and here we will look at four important methods: using a recursive equation, an explicit formula, a generating function, and an integral expression.

These numbers help mathematicians understand patterns in sums and series. They are connected to important functions and formulas, making them useful in advanced math studies.

Bernoulli numbers and the Riemann zeta function

The Bernoulli numbers are special numbers used in many areas of mathematics. They can be linked to the Riemann zeta function, which is an important tool for studying numbers. For example, there is a formula that connects Bernoulli numbers to the zeta function when certain conditions are met.

This connection helps mathematicians understand patterns in numbers better. When the zeta function is used with positive numbers, it creates relationships that show up in various mathematical problems.

Efficient computation of Bernoulli numbers

Mathematicians have developed clever ways to calculate Bernoulli numbers quickly. One method, called the "triangle algorithm," starts with a simple value and builds up using a pattern, much like Pascal's triangle. Another approach uses the tangent numbers and special formulas to find the Bernoulli numbers.

Researchers have created even faster ways to compute these numbers, especially when dealing with very large values. These methods help mathematicians solve complex problems and have been used to calculate Bernoulli numbers for extremely large n, such as 10⁸. These tools are built into software like SageMath.

Applications of the Bernoulli numbers

Bernoulli numbers are important in mathematics. They help us understand sums of powers and special functions. For example, they are used in the Euler–Maclaurin formula, which helps approximate sums by integrals.

They also appear in formulas for adding up powers of numbers, like finding the sum of the first few squares or cubes. This is known as Faulhaber's formula. Additionally, Bernoulli numbers show up in the expansions of trigonometric and hyperbolic functions, such as the tangent function.

Connections with combinatorial numbers

The Bernoulli numbers are closely tied to combinatorial mathematics. They connect with various combinatorial concepts through the inclusion–exclusion principle and other mathematical structures.

One key connection is through Worpitzky numbers, which involve factorials and powers. These numbers help express Bernoulli numbers as sums, showing how they emerge from combinatorial calculations. Similarly, Stirling numbers of the second kind link Bernoulli numbers to polynomial expressions and falling factorials, illustrating their role in expanding and simplifying mathematical series.

Bernoulli numbers also relate to Pascal’s triangle and Eulerian numbers, providing formulas that tie these triangular arrangements of numbers to deeper mathematical properties. These connections highlight how Bernoulli numbers appear across different areas of mathematics, especially in sequences and series.

A binary tree representation

The Bernoulli numbers can be calculated using a special kind of math tree, called a binary tree. This method was described by a mathematician named S. C. Woon. In this method, each node in the tree follows specific rules to create left and right branches. By adding up values from specific nodes on the tree, we can find the Bernoulli numbers.

For example, the first few Bernoulli numbers can be found using simple calculations from the tree nodes. This shows how patterns in numbers can be discovered through creative mathematical structures.

Integral representation and continuation

The Bernoulli numbers can be described using special mathematical expressions called integrals. These integrals connect the Bernoulli numbers to another important math concept called the Riemann zeta function.

For example, when certain values are plugged into these integral expressions, they match specific Bernoulli numbers. Mathematician Leonhard Euler studied these connections and calculated some interesting patterns, showing how the Bernoulli numbers appear in various mathematical series.

The relation to the Euler numbers and π

The Euler numbers are a group of numbers closely related to the Bernoulli numbers. They help us understand how these numbers connect to the number π, which many know as 3.14159....

One important idea is that the Euler numbers are roughly (2/π) times bigger than the Bernoulli numbers when certain conditions are met. This relationship shows that π is deeply connected to both the Bernoulli and Euler numbers.

Mathematicians have found special formulas that let them change Bernoulli numbers into Euler numbers and vice versa. These formulas show that both types of numbers share a common mathematical root, tied closely to π.

An algorithmic view: the Seidel triangle

The sequence ( S_n ) has an interesting property: the denominators of ( S_{n+1} ) divide the factorial ( n! ). This means the numbers ( T_n = S_{n+1} \cdot n! ), known as Euler zigzag numbers, are always whole numbers.

These numbers have a special connection to trigonometric functions. Their exponential generating function is the sum of the secant and tangent functions. This connection helps mathematicians rewrite Bernoulli and Euler numbers in terms of the sequence ( T_n ), making them easier to calculate.

A combinatorial view: alternating permutations

Main article: Alternating permutations

In the late 1800s, a mathematician named Désiré André discovered an interesting pattern while looking at the beginnings of some important math functions, tan x and sec x. He found that the numbers that appear in these patterns are related to something called Euler zigzag numbers.

André also showed that these special numbers can be used to count certain arrangements of objects, called alternating permutations, depending on whether the number of objects is odd or even. This connection helps mathematicians understand both the functions and the counting problem better.

Related sequences

The Bernoulli numbers are connected to other sequences of numbers through special mathematical methods. For example, by averaging the first two Bernoulli numbers, we get another set called associate Bernoulli numbers. These numbers help create patterns known as the Balmer series.

Using a method called the Akiyama–Tanigawa algorithm on other number sequences, we can generate the Bernoulli numbers or their variations, called intrinsic Bernoulli numbers. These connections show how deeply linked different number patterns can be in mathematics.

Arithmetical properties of the Bernoulli numbers

Bernoulli numbers are a special set of numbers that show up in many areas of mathematics. They help us understand patterns in sums and functions. For example, they are connected to the Riemann zeta function, which studies the distribution of prime numbers.

One important idea is that Bernoulli numbers relate to Fermat's Last Theorem, a famous problem about solving equations with powers. They also help us understand deep properties of prime numbers and how they behave in certain mathematical structures. These numbers have many surprising connections and uses in number theory.

Generalized Bernoulli numbers

The generalized Bernoulli numbers are special algebraic numbers that are connected to certain values of Dirichlet L-functions, much like regular Bernoulli numbers are connected to the Riemann zeta function. They are defined using a mathematical process that involves a Dirichlet character.

One important rule for these numbers is that, except for one special case, many of them equal zero. This helps mathematicians understand their patterns and uses. These numbers also help explain relationships in advanced math, similar to how regular Bernoulli numbers do.

Eisenstein–Kronecker number

Main article: Eisenstein–Kronecker number

Eisenstein–Kronecker numbers are similar to generalized Bernoulli numbers but are used with imaginary quadratic fields. They help study important values in math related to Hecke characters.

Appendix

Bernoulli numbers are a special set of numbers used in mathematics. They help solve problems involving sums of powers and appear in many important formulas. These numbers can also be found in the expansions of certain mathematical functions.