Consider a sequence of polynomials *P _{n}*(

*x*), where

*P*is an

_{n}*n*th degree polynomial. Let us also have the relationship that . Such a sequence is known as an Appell sequence, and includes a number of notable polynomial sequences (including the Hermite polynomials mentioned in a recent Physics Friday post). Note that the sequence

*P*(

_{n}*x*)=

*x*is a trivial example.

^{n}Now, we can obtain from the relation

, or taking the definite integral of both sides, .

Letting our integral limits become

*x*and

*x*+1, we have:

. Now, note that

*P*

_{n+1}(

*x*+1) is an

*n*+1 degree polynomial with the same coefficient on the

*x*

^{n+1}term as

*P*(

_{n}*x*), and thus must be a polynomial of degree no higher than

*n*; in fact, one can check and find it must be a degree

*n*polynomial.

So let us then choose the Appell sequence defined by

We can compute the first few terms immediately:

gives us

*B*

_{0}(

*x*)=1

Similarly, we can find

and so on.

Note that by putting

*x*=0 in our defining equation, we have

.

This sequence is known as the Bernoulli polynomials. Amongst its properties is the symmetry relation

,

which tells us that

.

By differentiating the defining equation with respect to x, we get:

Now, consider the integral , for

*m*a positive integer. Breaking up the region of integration into unit intervals, we find

and using our defining relation , we find that

And thus we see:

.

So

This is Faulhaber’s formula for the sum of the

*n*th power of the first

*m*integers in terms of the Bernoulli polynomials (see here).

The values of the Bernoulli polynomials at

*x*=0 (the constant term of the polynomial) are known as the Bernoulli numbers:

.

The first few values are

For odd

*n*>1,

*B*=0; is the only non-zero Bernoulli number for odd

_{n}*n*. The even Bernoulli numbers, as one may observe from the above examples, alternate in sign.

Now, we recall that

Thus the coefficient of the

*x*term in

^{k}*B*(

_{n}*x*) is times the coefficient of the

*x*

^{k-1}term of

*B*

_{n-1}(

*x*); thus the coefficient of

*x*in

*B*(

_{n}*x*) is equal to . Similarly, the coefficient of

*x*

^{2}in

*B*(

_{n}*x*) is equal to ; the

*x*

^{3}coefficient is , and so on.

Thus we see:

.

(proof by induction can verify this formula).

Therefore:

From this, and our symmetry, we find a recursive relation for the Bernoulli numbers:

,

which can be solved for

*B*in terms of

_{n}*B*

_{0},

*B*

_{1},

*B*

_{2}, …,

*B*

_{n-1}:

,

which with

*B*

_{0}=1, can also be used to define the Bernoulli sequence.

Tags: Appell Sequence, Bernoulli Numbers, Bernoulli Polynomials, Faulhaber's Formula, Math, Monday Math

January 30, 2009 at 11:13 pm |

So this is absolutely awesome. I’ve been doing random link walking (clicking an entry in a blog’s blogroll, click a random entry on the next blog’s blogroll, etc.) and found myself here. I had recently tried to work out a general formula for sum(x^n) and found that the sequence of equations contained these numbers that seemingly had no pattern. A little searching and I found they were Bernoulli numbers. Serendipity led me here.

This, by the way, if the first blog I’ve seen flush with equations. Are these images (e.g. from a word document or LaTex), or is the capability native to WordPress?

Thanks…I’ll be back to dig through old posts.

February 1, 2009 at 9:34 am |

Thanks, skepTick. No, the equations aren’t something native to WordPress. I use John Forkosh’s mimeTeX, and the public mimeTeX server.

February 2, 2009 at 4:13 am |

[…] Math 57 By twistedone151 Previously, we introduced the Bernoulli polynomials, and the Bernoulli numbers. In particular, we found that […]

February 2, 2009 at 5:37 pm |

Thank you. Since I asked the question, I did some searching and seems like there are several possibilities for use with blogger. Maybe one day blog hosts will give a little consideration to the mathematician in all of us.