# Equivalence of Definitions of Bernoulli Numbers

## Theorem

The following definitions of the concept of Bernoulli Numbers are equivalent:

### Generating Function

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

### Recurrence Relation

$B_n = \begin{cases} 1 & : n = 0 \\ \ds - \sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n + 1 - k} & : n > 0 \end{cases}$

or equivalently:

$B_n = \begin{cases} 1 & : n = 0 \\ \ds - \frac 1 {n + 1} \sum_{k \mathop = 0}^{n - 1} \binom {n + 1} k B_k & : n > 0 \end{cases}$

## Proof

From the generating function definition:

 $\ds \frac x {e^x - 1}$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!} \paren {\sum_{k \mathop = 0}^\infty \frac {x^k} {k!} - 1}$ Definition of Real Exponential Function $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!} \sum_{k \mathop = 0}^\infty \frac {x^{k + 1} } {\paren {k + 1}!}$ $1 = \dfrac {x^0} {0!}$ $\ds \leadsto \ \$ $\ds 1$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!} \sum_{k \mathop = 0}^\infty \frac {x^k} {\paren {k + 1}!}$ $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \sum_{k \mathop = 0}^n \binom n k \frac {B_k} {n - k + 1}$ Cauchy Product, Product of Absolutely Convergent Series

Equating coefficients:

For $n = 0$:

 $\ds 1$ $=$ $\ds \sum_{k \mathop = 0}^0 \binom 0 k \frac {B_k} {0 - k + 1}$ $\ds$ $=$ $\ds \binom 0 0 \frac {B_0} {0 - 0 + 1}$ $\ds$ $=$ $\ds B_0$ Binomial Coefficient with Zero

For $n > 0$:

 $\ds 0$ $=$ $\ds \frac 1 {n!} \sum_{k \mathop = 0}^n \binom n k \frac {B_k} {n - k + 1}$ $\ds$ $=$ $\ds \frac 1 {n!} \paren {\sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n - k + 1} + \binom n n \frac {B_n} {n - n + 1} }$ $\ds$ $=$ $\ds \sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n - k + 1} + B_n$ Binomial Coefficient with Self and simplifying $\ds \leadsto \ \$ $\ds B_n$ $=$ $\ds -\sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n - k + 1}$

Hence the result:

$B_n = \begin{cases} 1 & : n = 0 \\ \ds - \sum_{k \mathop = 0}^{n-1} \binom n k \frac {B_k} {n - k + 1} & : n > 0 \end{cases}$

$\blacksquare$