Definition:Bernoulli Numbers/Generating Function

Definition

The Bernoulli numbers $B_n$ are a sequence of rational numbers defined by the exponential generating function:

$\ds \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$

Also see

• Equivalence of Definitions of Bernoulli Numbers

• Results about the Bernoulli Numbers can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Bernoulli number
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler
• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: $(29)$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Bernoulli numbers
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Bernoulli numbers
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Bernoulli number

• Weisstein, Eric W. "Bernoulli Numbers." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BernoulliNumber.html