Tuesday, 17 June 2014

Bernoulli Numbers and Polynomials

Bernoulli Numbers and Polynomials

The sum of first n natural numbers 1, 2, 3,,n is

S1(n) := m=1nm = n(n + 1) 2 = n2 2 + n 2.

This formula can be derived by noting that

S1(n) = 1 + 2 + + n S1(n) = n + (n 1) + + 1.

Therefore, summing term-by-term,

2S1(n) = (n + 1) + + (n + 1) ntimes = n(n + 1).

An alternate way of obtaining the above sum is by using the following two identity:

(i)
(m + 1)2 m2 = 2m + 1 and, hence,
2 m=1nm = m=1n (m + 1)2 m2 n.

(ii)
m=1n (m + 1)2 m2 = [22 12] + [32 22] + + +[n2 (n 1)2] + [(n + 1)2 n2] = (n + 1)2 1.

Thus,

S1(n) = (n + 1)2 (n + 1) 2 = n(n + 1) 2 .

More generally, the sum of k-th power of first n natural numbers is denoted as

Sk(n) := 1k + 2k + + nk.

Since a0 = 1, for any a, we have S0(n) = n. For k , one can compute Sk(n) using the identities:

(i)
(m + 1)k+1 mk+1 = i=0kk + 1 i mi

and, hence,

(k+1) m=1nmk = m=1n (m + 1)k+1 mk+1 i=0k1 m=1nk + 1 i mi.

(ii)
m=1n (m + 1)k+1 mk+1 = (n + 1)k+1 1.

Thus,

(k + 1) m=1nmk = (n + 1)k+1 1 i=0k1k + 1 i m=1nmi Sk(n) = (n + 1)k+1 k + 1 (n + 1) k + 1 1 k + 1 i=1k1k + 1 i Si(n).

The formula obtained in RHS is a (k + 1)-degree polynomial of n. Using the above formula, one can compute

S2(n) = n3 3 + n2 2 + n 6, S3(n) = n4 4 + n3 2 + n2 4 , S4(n) = n5 5 + n4 2 + n3 3 n 30, S5(n) = n6 6 + n5 2 + 5n4 12 n2 12, S6(n) = n7 7 + n6 2 + n5 2 n3 6 + n 42, S7(n) = n8 8 + n7 2 + 7n6 12 7n4 24 + n2 12,

James/Jacques/Jakob Bernoulli observed that the sum of first n whole numbers raised to the k-th power can be concisely written as,

Sk(n) = nk+1 k + 1 + 1 2nk + 1 12knk1 + 0 × nk2 + .

Note that the coefficients 1, 12, 112, 0, are independent of k. Jakob Bernoulli rewrote the above expression as

(k + 1)Sk(n) = nk+1 1 2(k + 1)nk + 1 6 k(k + 1) 2 nk1 + 0 1 30 (k 2)(k 1)k(k + 1) 4! +

and, hence,

Sk(n) = 1 k + 1 i=0kk + 1 i Bink+1i, (1)

where Bi are the i-th Bernoulli numbers and the formula is called Bernoulli formula. An easier way to grasp the above formula for Sk(n) is to rewrite1 it as

Sk(n) = (n + B)k+1 B k+1 k + 1

where B is a notation used to identify the i-th power of B with the i-th Bernoulli number Bi and

(n + B)k+1 := i=0k+1k + 1 i Bink+1i.

This notation also motivates the definition of Bernoulli polynomial of degree k as

Bk(t) := i=0kk i Bitki

where Bi are the Bernoulli numbers. In terms of Bernoulli polynomials, the k-th Bernoulli number Bk = Bk(0).

Two quick observation can be made from (1).

(i)
There is no constant term in Sk(n) because i does not take k + 1.
(ii)
The k-th Bernoulli number, Bk, is the coefficient of n in Sk(n). For instance, B0 is coefficient of n in S0(n) = n and, hence, B0 = 1. Similarly, B1 = 12,B2 = 16,B3 = 0,B4 = 130,B5 = 0,B6 = 142,B7 = 0,.

The beauty about the sequence of Bernoulli numbers is that one can compute them a priori and use it to calculate Sk(n), i.e., given n and k {0} it is enough to know Bi, for all 0 i k, to compute Sk(n). We already computed B0 = 1. Using n = 1 in the Bernoulli formula (1), we get

1 = 1 k + 1 i=0kk + 1 i Bi (2)

and, this implies that the k-th Bernoulli number, for k > 0, is defined as

Bk = 1 1 k + 1 i=0k1k + 1 i Bi.

Recall that B3,B5,B7 vanish. In fact, it turns out that Bk = 0 for all odd k > 1. The odd indexed Bernoulli numbers vanish because they have no n-term. Since there are no constant terms in Sk(n), the vanishing of Bernoulli numbers is equivalent to the fact that n2 is a factor of Sk(n). Some properties of Bernoulli numbers:

(i)
For odd k > 1, Bk = 0.
(ii)
For even k, Bk0.
(iii)
Bk for all k {0}.
(iv)
B0 = 1 is the only non-zero integer.
(v)
B4j is negative rational and B4j2 is positive rational, for all j .
(vi)
|B6 = 142| < |B2k|, for all k .

L. Euler gave a nice generating function for the Bernoulli numbers. He seeked a function f(x) such that f(k)(0) = B k where f(k) denotes the k-th derivative of f with the convention that f(0) = f. If such an f exists then it admits the Taylor series expansion, around 0,

f(x) = k=0f(k)(0)xk k! .

Therefore, for such a function

f(x) = k=0B kxk k! .

Recall the Taylor series of ex,

ex = k=0xk k!

defined for all x . Consider the product (discrete convolution/Cauchy product) of the infinite series

f(x)ex = k=0B kxk k! k=0xk k! = k=0 i=0kB ixi i! xki (k i)! = k=0 i=0k Bi i!(k i)! xk = k=0 i=0kk i Bi xk k! .

Let

ck := i=0kk i Bi = i=0k1k i Bi + Bk = k + Bk.

Then

f(x)ex = k=0(k + B k)xk k! = k=1 xk (k 1)! + f(x) = xex + f(x).

Thus, the f we seek satisfies

f(x) = xex ex 1

and is called the generating function. Since ex > 0 for all x , we may rewrite f(x) as

f(x) = x 1 ex. (3)

The entire exercise of seeking f can be generalised to complex numbers and

f(z) = z 1 ezz  with 0 (z) < 2π.

A word of caution that idenitities (2) and (3) are different from the standard formulae because we have derived them for second Bernoulli numbers, viz., with B1 = 12. The standard convention is to work with first Bernoulli numbers, viz., with B1 = 12. The first Bernoulli numbers can be obtained by following the approach of summing the k-th powers of first n 1 natural numbers, for any given n.

The Bernoulli numbers with appeared while computing Sk(n) is appears in many crucial places.

(a)
In the expansion of tan z.
tan z = k=1(1)k122k(22k 1)B 2k 2n! z2k1.

for all |z| < π2.

(b)
In computing the sum of Riemann-zeta function
ζ(s) = n=1 1 ns

for positive even integers s. The case s = 2 is the famous Basel problem computed by Euler to be π26.

Theorem 1 (Euler). For all k ,

ζ(2k) = (1)k+1(2π)2k 2(2k)!B2k.

Further, the relation ζ(2k) = B2k+1 2k+1 gives the trivial zeroes of the Riemann zeta function.

(c)
The Bernoulli numbers were also used as an attempt to prove Fermat’s last theorem (already discussed in a previous article/blog).

Definition 1. An odd prime number p is called regular if p does not divide the numerator of Bk, for all even k p 3. Any odd prime which is not regular is called irregular.

The odd primes 3, 5, 7,, 31 are all regular primes. The first irregular prime is 37. It is an open question: are there infinitely many regular primes? However, it is known that there are infinitely many irregular primes.

Theorem 2 (Kummer, 1850). If p is a regular odd prime then the equation

ap + bp = cp

has no solution in .