Email Gateway (Aug 22 2022 at 14:59): Email Gateway (Aug 22 2022 at 14:59):From: Tobias Nipkow <nipkow@in.tum.de>
Bernoulli Numbers
Lukas Bulwahn and Manuel Eberl
Bernoulli numbers were first discovered in the closed-form expansion of the sum
1^m + 2^m + … + n^m for a fixed m and appear in many other places. This entry
provides three different definitions for them: a recursive one, an explicit one,
and one through their exponential generating function.
In addition, we prove some basic facts, e.g. their relation to sums of powers of
integers and that all odd Bernoulli numbers except the first are zero. We also
prove the correctness of the Akiyama–Tanigawa algorithm for computing Bernoulli
numbers with reasonable efficiency, and we define the periodic Bernoulli
polynomials (which appear e.g. in the Euler–MacLaurin summation formula and the
expansion of the log-Gamma function) and prove their basic properties.
https://www.isa-afp.org/entries/Bernoulli.shtml
Enjoy!
smime.p7s
Last updated: Nov 21 2024 at 12:39 UTC