Bernoulli number¶

Define a rational-number sequence by the exponential generating function for x divided by exp(x) minus one, yielding coefficients that govern power sums, Euler-Maclaurin correction, and zeta-value identities.

Version: v3 · 2026-09-19 · History
Domain-specific #: 1369
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Number Theory and Analysis, Special Number Sequences and Generating Functions → Mathematics

Core Idea¶

Under the modern first convention, the Bernoulli numbers are the rational coefficients determined by \(\frac{x}{e^x-1}=\sum_{n=0}^{\infty}B_n\frac{x^n}{n!}\), so \(B_0=1\), \(B_1=-\tfrac12\), and \(B_{2m+1}=0\) for \(m\geq1\). Multiplying the defining formal series by the series for the exponential minus one and equating coefficients yields a triangular recurrence, so each new value is fixed by earlier values; the same coefficient family enters Bernoulli polynomials and correction terms for replacing sums by integrals.

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The Recipe Number List

Bernoulli numbers are a special list of fractions that come from one fixed math recipe. Each new number in the list can be worked out from the ones before it. They are useful for shortcuts, like quickly adding up 1 + 4 + 9 + 16 and so on, without adding every number one at a time.

Numbers From a Special Recipe

The Bernoulli numbers are a famous list of numbers, mostly fractions, written B0, B1, B2 and so on. They come from one exact recipe, and each new one can be figured out from the ones before it. With the usual choice, B0 is 1, B1 is −1/2, and every odd-numbered one after that is 0. Mathematicians use them for things like quick formulas for adding up powers, such as 1² + 2² + 3² + …. Because there are a couple of versions of the recipe, you always have to say which one you're using, especially for B1.

Series Coefficients of x/(eˣ−1)

The Bernoulli numbers B_n are the rational numbers defined as the coefficients in the expansion x/(e^x − 1) = Σ B_n x^n/n!. Under this convention B_0 = 1, B_1 = −1/2, and B_n = 0 for every odd n greater than 1. Multiplying both sides by the series for e^x − 1 and matching coefficients gives a recurrence, so each value is determined by the earlier ones. They show up in formulas for sums of powers, in the Euler-Maclaurin formula that corrects the error when a sum is replaced by an integral, and in values of the Riemann zeta function at even integers. Care is needed: some sources use B_1 = +1/2, and the n! in the formula (an exponential generating function) matters.

Under the modern convention, the Bernoulli numbers are the rational coefficients Bₙ in the exponential generating function x/(eˣ − 1) = Σₙ Bₙ xⁿ/n!, giving B₀ = 1, B₁ = −1/2, and B₂ₘ₊₁ = 0 for m ≥ 1. Multiplying the series by eˣ − 1 and equating coefficients yields a triangular recurrence, so each Bₙ is determined by its predecessors. The same coefficients appear in Bernoulli polynomials and in the correction terms of the Euler–Maclaurin formula for approximating sums by integrals. Applications include closed forms for sums of powers, relations between even-index Bernoulli numbers and values of the Riemann zeta function, and the arithmetic of their denominators. Correct use depends on normalization: the generating function is exponential, not ordinary, so the factorial must be kept; the sign of B₁ differs between conventions and must not be mixed within one identity; odd-index vanishing does not apply at index one; and the definition is a formal-series identity, not a convergence claim. The Bernoulli numbers should not be confused with Bernoulli trials, Bernoulli polynomials as functions, or Euler numbers.

Scope of Application¶

Bernoulli number applies when the analyst can specify a sequence of rational numbers \((B_n)_{n\geq0}\) interpreted as coefficients of a formal or analytic exponential generating function and establish that one declared generating-function or equivalent recurrence convention uniquely fixes every rational coefficient and explicitly resolves the sign ambiguity at index one. The entry states exact mathematical definitions and identities. Numerical computation, asymptotic truncation, p-adic interpolation, generalized characters, and software conventions require their own error and normalization controls.

Clarity¶

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because Bernoulli number can refer to either first-index sign convention, while Bernoulli also names a distribution, process, trial, polynomial family, and many generalized sequences. The disciplined statement is that the object counts as Bernoulli number exactly when one declared generating-function or equivalent recurrence convention uniquely fixes every rational coefficient and explicitly resolves the sign ambiguity at index one

Manages Complexity¶

The abstraction compresses first and second conventions, Bernoulli polynomials, generalized and higher-order numbers, recurrence and explicit formulas, zeta-value expressions, denominator theorems, and computational representations into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares index, first-number sign, generating-function normalization, formal or analytic reading, recurrence form, polynomial specialization, denominator, parity, zeta identity, and generalized parameter and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning¶

Type the carrier. Establish a sequence of rational numbers \((B_n)_{n\geq0}\) interpreted as coefficients of a formal or analytic exponential generating function and reject examples from a different problem.
Lock the rule. Express that one declared generating-function or equivalent recurrence convention uniquely fixes every rational coefficient and explicitly resolves the sign ambiguity at index one independently of one notation or implementation.

Knowledge Transfer¶

Transfer within number theory and analysis is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from Coefficient comparison gives \(B_0=1\), \(B_1=-\tfrac12\), \(B_2=\tfrac16\), and \(B_4=-\tfrac1{30}\), while the odd values beyond \(B_1\) vanish. to Euler-Maclaurin summation uses even Bernoulli numbers as coefficients of endpoint-derivative corrections when approximating a finite sum by an integral. demonstrates that continuity.

Relationships to Other Abstractions¶

Current abstraction Bernoulli number Domain-specific

Parents (1) — outgoing typed relations

Bernoulli number is a kind of Recurrence Prime

The defining coefficient identity yields a literal triangular dependence of each indexed value on prior values, instantiating recurrence; rationality, normalization, and analytic-number-theory identities provide the specialization.

Hierarchy path (1) — routes to 1 parentless root

Bernoulli number → is a kind of Recurrence

Neighborhood in Abstraction Space¶

Bernoulli number sits in a crowded region of the domain-specific corpus (30^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25