Generating function for the Bernoulli Polynomials

A generating function is a power series whose coefficients are a sequence of interest. The Bernoulli Polynomials have the generating function

$\displaystyle \frac{te^{xt}}{e^{t}-1}=\sum ^{\infty}_{n=0}B_{n}(x) \frac{t^n}{n!}$

at least when |t| < 2π

The derivation of this result is shown here.

A generating function often enables properties of the sequence to be easily exposed. Here the generating function enables us to show simply that the odd Bernoulli numbers B_2k+1 are zero (k > 0). This is necessarily the case if

$\displaystyle \frac{t}{e^{t}-1}-B_{1}t=1+\sum ^{\infty}_{n=2}B_{n} \frac{t^n}{n!}$

is an even function of t. This is so since

$\displaystyle \frac{t}{e^{t}-1}+\frac{t}{2}=\frac{t(e^{t}+1)}{2(e^{t}-1)}=\frac{-t(e^{-t}+1)}{2(e^{-t}-1)}=\frac{-t}{e^{-t}-1}-\frac{t}{2}$

Tags: Bernoulli Numbers, Riemann Zeta Function

This entry was posted on November 22, 2011 at 11:00 am and is filed under Finite Differences. You can follow any responses to this entry through the RSS 2.0 feed. You can skip to the end and leave a response. Pinging is currently not allowed.

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Some Classical Maths

Generating function for the Bernoulli Polynomials

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