A General Lacunary Recurrence Formula

Abstract

The Bernoulli numbers B _n may be defined by means of the generating function

$$ \frac{x}{{{e^{^x}} - 1}} = \sum\limits_{n = 0}^\infty {{B_n}} \frac{{{x^n}}}{{n!}} $$

((1.1))

. An example of a “lacuary” recurrence for these numbers is

$$ \sum\limits_{j = 0}^n {\left( \begin{array}{l}6n + 3 \\6j \\\end{array} \right)} {B_{6j}} = 2n + 1 $$

((1.2))

. This recurrence has lacunae, or gaps, or length 6. That is, to compute B _6n , it is not necessary to know the values of B _j for all j < 6n; we need only know the values of B _6j for j = 0, 1, ..., n - 1.