# Thread: Proving a Bernoulli Relationship

1. ## Proving a Bernoulli Relationship

I have already managed to prove the relationship

By considering the integral

I am now trying to prove

This is what I have managed

to do so far but now I have no idea where to move forward

2. ## Re: Proving a Bernoulli Relationship

Originally Posted by klw289
I have already managed to prove the relationship

By considering the integral

I am now trying to prove

This is what I have managed

to do so far but now I have no idea where to move forward
You can start from the 'explicit definition' of the 'Bernoulli polynomial of index n'...

$B_{n}(x)= \sum_{k=0}^{n} \binom{n}{k}\ B_{k}\ x^{n-k}$ (1)

... where $B_{k}$ isw the 'Bernoulli number of index k'. Deriving (1) You obtain...

$B_{n}^{'}(x)= \sum_{k=0}^{n} \binom{n}{k}\ B_{k}\ (n-k)\ x^{n-k-1}=$

$= n\ \sum_{k=0}^{n-1} \binom{n-1}{k}\ B_{k}\ x^{n-k-1} = n\ B_{n-1}(x)$ (2)

Kind regards

$\chi$ $\sigma$

3. ## Re: Proving a Bernoulli Relationship

I've have already proved that relationship as I mentioned, I used it to help me integrate the intergral I was asked to consider but how do I use these results to get to the proof for Bn+1 which I am looking for