Monday Math 56

Consider a sequence of polynomials Pn(x), where Pn is an nth 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 Pn(x)=xn is a trivial example.

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 Pn+1(x+1) is an n+1 degree polynomial with the same coefficient on the xn+1 term as Pn(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
B0(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 nth 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, Bn=0; is the only non-zero Bernoulli number for odd 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 xk term in Bn(x) is times the coefficient of the xk-1 term of Bn-1(x); thus the coefficient of x in Bn(x) is equal to . Similarly, the coefficient of x2 in Bn(x) is equal to ; the x3 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 Bn in terms of B0, B1, B2, …, Bn-1:
,
which with B0=1, can also be used to define the Bernoulli sequence.

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4 Responses to “Monday Math 56”

  1. The skepTick Says:

    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.

  2. twistedone151 Says:

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

  3. Monday Math 57 « Twisted One 151’s Weblog Says:

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

  4. The skepTick Says:

    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.

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