Different Viewpoints and Conventions
The Bernoulli numbers can be regarded from four main viewpoints:
- as standalone arithmetical objects,
- as combinatorial objects,
- as values of a sequence of certain polynomials,
- as values of the Riemann zeta function.
Each of these viewpoints leads to a set of more or less different conventions.
- Bernoulli numbers as standalone arithmetical objects.
Associated sequence: 1/6, −1/30, 1/42, −1/30, …
This is the viewpoint of Jakob Bernoulli. (See the cutout from his Ars Conjectandi, first edition, 1713). The Bernoulli numbers are understood as numbers, recursive in nature, invented to solve a certain arithmetical problem, the summation of powers, which is the paradigmatic application of the Bernoulli numbers. These are also the numbers appearing in the Taylor series expansion of tan(x) and tanh(x). It is misleading to call this viewpoint 'archaic'. For example Jean-Pierre Serre uses it in his highly acclaimed book A Course in Arithmetic which is a standard textbook used at many universities today.
- Bernoulli numbers as combinatorial objects.
Associated sequence: 1, +1/2, 1/6, 0, …
This view focuses on the connection between Stirling numbers and Bernoulli numbers and arises naturally in the calculus of finite differences. In its most general and compact form this connection is summarized by the definition of the Stirling polynomials σn(x), formula (6.52) in Concrete Mathematics by Graham, Knuth and Patashnik.
In consequence Bn = n! σn(1) for n ≥ 0.
- Bernoulli numbers as values of a sequence of certain polynomials.
Assuming the Bernoulli polynomials as already introduced the Bernoulli numbers can be defined in two different ways:
- Bn = Bn(0). Associated sequence: 1, −1/2, 1/6, 0, …
- Bn = Bn(1). Associated sequence: 1, +1/2, 1/6, 0, …
The two definitions differ only in the sign of B1. The choice Bn = Bn(0) is the convention used in the Handbook of Mathematical Functions.
- Bernoulli numbers as values of the Riemann zeta function.
Associated sequence: 1, +1/2, 1/6, 0, …
Using this convention, the values of the Riemann zeta function satisfy nζ(1 − n) = −Bn for all integers n≥0. (See the paper of S. C. Woon; the expression nζ(1 − n) for n = 0 is to be understood as limx → 0 xζ(1 − x).)
Read more about this topic: Bernoulli Number
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