The sum of ﬁrst natural numbers is
This formula can be derived by noting that
Therefore, summing term-by-term,
An alternate way of obtaining the above sum is by using the following two identity:
More generally, the sum of -th power of ﬁrst natural numbers is denoted as
Since , for any , we have . For , one can compute using the identities:
The formula obtained in RHS is a -degree polynomial of . Using the above formula, one can compute
James/Jacques/Jakob Bernoulli observed that the sum of ﬁrst whole numbers raised to the -th power can be concisely written as,
Note that the coeﬃcients are independent of . Jakob Bernoulli rewrote the above expression as
where are the -th Bernoulli numbers and the formula is called Bernoulli formula. An easier way to grasp the above formula for is to rewrite1 it as
where is a notation used to identify the -th power of with the -th Bernoulli number and
This notation also motivates the deﬁnition of Bernoulli polynomial of degree as
where are the Bernoulli numbers. In terms of Bernoulli polynomials, the -th Bernoulli number .
Two quick observation can be made from (1).
- There is no constant term in because does not take .
- The -th Bernoulli number, , is the coeﬃcient of in . For instance, is coeﬃcient of in and, hence, . Similarly, .
The beauty about the sequence of Bernoulli numbers is that one can compute them a priori and use it to calculate , i.e., given and it is enough to know , for all , to compute . We already computed . Using in the Bernoulli formula (1), we get
and, this implies that the -th Bernoulli number, for , is deﬁned as
Recall that vanish. In fact, it turns out that for all odd . The odd indexed Bernoulli numbers vanish because they have no -term. Since there are no constant terms in , the vanishing of Bernoulli numbers is equivalent to the fact that is a factor of . Some properties of Bernoulli numbers:
- For odd , .
- For even , .
- for all .
- is the only non-zero integer.
- is negative rational and is positive rational, for all .
- , for all .
L. Euler gave a nice generating function for the Bernoulli numbers. He seeked a function such that where denotes the -th derivative of with the convention that . If such an exists then it admits the Taylor series expansion, around ,
Therefore, for such a function
Recall the Taylor series of ,
deﬁned for all . Consider the product (discrete convolution/Cauchy product) of the inﬁnite series
Thus, the we seek satisﬁes
and is called the generating function. Since for all , we may rewrite as
The entire exercise of seeking can be generalised to complex numbers and
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 . The standard convention is to work with ﬁrst Bernoulli numbers, viz., with . The ﬁrst Bernoulli numbers can be obtained by following the approach of summing the -th powers of ﬁrst natural numbers, for any given .
The Bernoulli numbers with appeared while computing is appears in many crucial places.
- In the expansion of .
for all .
- In computing the sum of Riemann-zeta function
for positive even integers . The case is the famous Basel problem computed by Euler to be .
Further, the relation gives the trivial zeroes of the Riemann zeta function.
- The Bernoulli numbers were also used as an attempt to prove Fermat’s last
theorem (already discussed in a previous article/blog).
The odd primes are all regular primes. The ﬁrst irregular prime is . It is an open question: are there inﬁnitely many regular primes? However, it is known that there are inﬁnitely many irregular primes.