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

v2 · 2026-08-30 · History

Domain-specific #

1369

Origin domain

number theory and analysis

Subdomain

special number sequences and generating functions

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.

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

1. 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. 2. 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) — more general patterns this builds on

• Bernoulli number is a kind of Recurrence Prime

  The proposed strict upward parent is prime:recurrence.

Hierarchy path (1) — routes to 1 parentless root

• Bernoulli number → Recurrence

Neighborhood in Abstraction Space

Bernoulli number sits in a moderately populated region (49^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Special Polynomial Sequences & Identities (6 abstractions)

Nearest neighbors

• E-function — 0.90
• Mott polynomials — 0.89
• Poly-Bernoulli number — 0.88
• Faulhaber's formula — 0.88
• Constant-recursive sequence — 0.88

Computed from structural-signature embeddings · 2026-09-08