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Concept
Bernoulli number
Summary
In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.
The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by and ; they differ only for n = 1, where and . For every odd n > 1, Bn = 0. For every even n > 0, Bn is negative if n is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials , with and .
The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Takakazu. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.
The superscript ± used in this article distinguishes the two sign conventions for Bernoulli numbers. Only the n = 1 term is affected:
B with B = −1/2 ( / ) is the sign convention prescribed by NIST and most modern textbooks.
B with B = +1/2 ( / ) was used in the older literature, and (since 2022) by Donald Knuth following Peter Luschny's "Bernoulli Manifesto".
In the formulas below, one can switch from one sign convention to the other with the relation , or for integer n = 2 or greater, simply ignore it.
Since B_n = 0 for all odd n > 1, and many formulas only involve even-index Bernoulli numbers, a few authors write "B_n" instead of B_2n .
Official source
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