Bernoulli Number

Bernoulli Number

In mathematics, the Bernoulli numbers B_n are a sequence of rational numbers with deep connections to number theory. The values of the first few Bernoulli numbers are

B₀ = 1, B₁ = ±1⁄₂, B₂ = 1⁄₆, B₃ = 0, B₄ = −1⁄₃₀, B₅ = 0, B₆ = 1⁄₄₂, B₇ = 0, B₈ = −1⁄₃₀.

If the convention B₁=−1⁄₂ is used, this sequence is also known as the first Bernoulli numbers (A027641 / A027642 in OEIS); with the convention B₁=+1⁄₂ is known as the second Bernoulli numbers (A164555 / A027642 in OEIS). Except for this one difference, the first and second Bernoulli numbers agree. Since B_n=0 for all odd n>1, and many formulas only involve even-index Bernoulli numbers, some authors write B_n instead of B_2n.

The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery was posthumously published in 1712 in his work Katsuyo Sampo; 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 computer program.