Bernoulli number

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

The first Bernoulli numbers have specific values and 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 math 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.

Notation

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

• 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, the difference does not matter. 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 special numbers used 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 many math formulas.

The idea came from the need to find quick ways to add up powers of numbers, like adding the squares or cubes. Early mathematicians such as Pythagoras and Archimedes studied these sums, but Jacob Bernoulli found a unified way to handle them using what we now call Bernoulli numbers. His work gave a formula that works for all such sums, making calculations easier.

Definitions

Bernoulli numbers are special numbers used in mathematics. They appear in many different areas of math. There are several ways to describe them. We will look at four important methods:

• a recursive equation
• an explicit formula
• a generating function
• an integral expression

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

Bernoulli numbers and the Riemann zeta function

Bernoulli numbers are special numbers used in many parts of mathematics. They are linked to the Riemann zeta function, an important tool for studying numbers. There is a formula that connects Bernoulli numbers to the zeta function under certain conditions.

This connection helps mathematicians understand number patterns better. When the zeta function is used with positive numbers, it creates relationships that appear in many math problems.

Efficient computation of Bernoulli numbers

Mathematicians have found smart 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 made even faster ways to compute these numbers, especially for very large values. These methods help mathematicians solve tricky 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. This formula helps us estimate sums using integrals.

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

Connections with combinatorial numbers

Bernoulli numbers are linked to combinatorial mathematics. They connect with many ideas through the inclusion–exclusion principle and other math structures.

One key link is through Worpitzky numbers. These numbers use factorials and powers. They help show Bernoulli numbers as sums. This shows how they come from combinatorial math. Stirling numbers of the second kind also link Bernoulli numbers to polynomial expressions and falling factorials. This shows their role in expanding and simplifying math series.

Bernoulli numbers also relate to Pascal’s triangle and Eulerian numbers. They provide formulas that connect these number patterns to deeper math properties. These links show how Bernoulli numbers appear in different parts of math, especially in sequences and series.

A binary tree representation

Bernoulli numbers can be found using a special kind of math tree, called a binary tree. A mathematician named S. C. Woon described this method. In this method, each part of the tree follows rules to create branches to the left and right. By adding up values from some parts of the tree, we can calculate the Bernoulli numbers.

For example, the first few Bernoulli numbers can be found with easy math from the tree parts. This shows how number patterns can be discovered using creative math ideas.

Integral representation and continuation

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

When certain values are used in these integral expressions, they match specific Bernoulli numbers. Mathematician Leonhard Euler studied these connections and found interesting patterns. He showed how the Bernoulli numbers appear in many different math series.

The relation to the Euler numbers and π

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

One important idea is that the Euler numbers are about (2/π) times bigger than the Bernoulli numbers under certain conditions. This relationship shows that π is linked to both the Bernoulli and Euler numbers.

Mathematicians have found special formulas that let them change Bernoulli numbers into Euler numbers and back again. 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! ), called Euler zigzag numbers, are always whole numbers.

These numbers connect to trigonometric functions. Their exponential generating function is the sum of the secant and tangent functions. This helps mathematicians rewrite Bernoulli and Euler numbers using 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 found an interesting pattern in some important math functions, tan x and sec x. He discovered that the numbers in these patterns are related to something called Euler zigzag numbers.

André also showed that these special numbers can help count certain arrangements of objects, called alternating permutations, no matter if the number of objects is odd or even. This helps mathematicians understand both the functions and the counting problem better.

Related sequences

Bernoulli numbers are linked to other number patterns through special math rules. For example, by averaging the first two Bernoulli numbers, we get another set called associate Bernoulli numbers. These numbers help create patterns called the Balmer series.

Using a method called the Akiyama–Tanigawa algorithm on other number patterns, we can create Bernoulli numbers or their versions, called intrinsic Bernoulli numbers. These links show how different number patterns are connected in math.

Arithmetical properties of the Bernoulli numbers

Bernoulli numbers are special numbers that appear 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 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 connect to certain values of Dirichlet L-functions, just like regular Bernoulli numbers connect to the Riemann zeta function. They are defined using a mathematical process that includes a Dirichlet character.

One important rule for these numbers is that, except for one special case, many of them are zero. This helps mathematicians see patterns and understand how these numbers are useful. These numbers also help explain relationships in advanced math, much like 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 special numbers used in mathematics. They help solve problems with sums of powers and appear in many important formulas. You can also find these numbers when expanding some math functions.