Rapid calculation of and new recurrences for Bernoulli numbers, Euler numbers and other Rational Poly-exponential functions.

Abstract.

Bernoulli numbers and similar arithmetic objects have long been of interest in mathematics. Historically, people have been interested in different recursion formulae that can be derived for the Bernoulli numbers, and the use of these recursion formulae for the calculation of Bernoulli numbers. Some of these methods, which in the past have only been of theoretical interest, are now practical with the availability of high-powered computation.

This project explores some of these techniques of deriving new recursion formulae, and expands upon these methods. The main technique that is explored is that of ``multisectioning''. Typically, the calculation of a Bernoulli number requires the calculation of all previous Bernoulli numbers. The method of multisectioning is such that only a fraction of the previous Bernoulli numbers are needed. In exchange, a more complicated recursion formula, called a ``lacunary recursion formula'' must be derived and used.

This page contains some Maple code, help files, and worksheets to demonstrate how to perform these recurrences.

Maple libraries.

1. April 13, 1999, maple.lib, Maple Library, Version 1.0,
2. April 13, 1999, maple.ind Maple Index, Version 1.0,
3. April 13, 1999, maple.hdb Maple Help Database, Version 1.0,

Maple worksheets.

List of worksheets demonstrating this code