8. Bernoulli's and Euler's numbers

Bernoulli's numbers.

The function


isn't defined at x=0, but this singularity is removable because the limit as x→0 is finite, as we can easily see by applying De L'H˘pital's rule. The same thing is true for its derivatives:


We can therefore expand this function using the MacLaurin series expansion


where Bn is the limit as x→0 of the n-th derivative of f(x):


In this way we can find:




and so on. The Bn are said Bernoulli numbers, from the name of the mathematician who first used them in his Ars conjectandi, a treatise on probability theory.



The calculation of the derivatives and of their limits, even using powerful software, is slow and does'nt give any hints to conjecture a general relation between the numbers Bn which could allow a more direct algorithm to generate these numbers. Such algorithm can be found using the well known series expansion of the natural exponential function in the following way.

From Eqn3.gif we have


Now we multiplicate all the addends in the second sum by every addend of the first


The first and the second term are identical only if, in the second term, B0=1 and the sum of the coefficients of the same power of x is 0. So


The denominators in these sums coincide with the denominators of the factorial expansions of the binomial coefficients.


Using the binomial coefficients, we can write the n-th calculated row as


If we consider, for example, the last calculated row, we have


So, in general, we can write


This allows the recursive calculation of every Bn


To obtain further Bn, we may automatize the calculation as in the following Javascript application. If your browser does not allow the iframe tag, you can open the source page.

Bernoulli's numbers in WolframAlpha

From the explicitly calculated values and using the applet, we can conjecture that all the Barnoulli's numbers with odd index ≥ 3 are null. This is true, because from


we have


and after


So, every B in the first term must be 0.

Euler's numbers.

Given the function


let En be the values taken by it and its successive derivatives for x=0. The values En are said Euler's numbers. If we expand this function in Maclaurin series, we have


and then


If we explicitly represent some of the summands of the factors in the right-hand side, we obtain


The expansion of this product is




Two polynomials in the same variable are identical only if the same powers of the variable have equal coefficients, so the coefficients of powers of x in the right-hand side must coincide with the coefficients of the series expansion of the exponential. These identities allow us to calculate recursively the values of En. Infact


Obviously E5, like E1 and E3, is 0, because all En with odd n are null, since s(x) is an even function. The identity between the coefficients of the fifth power can be useful in conjecturing a general calculation method.


We can conjecture that, in general,


To obtain further En, we may automatize the calculation as in the following Javascript application. If your browser does not allow the iframe tag, you can open the source page.

There are different conventions on the notation of Euler numbers. Here we assume the convention adopted by Mathematica for the function EulerE[n].

Euler's numbers in WolframAlpha