Bernoulli polynomials (MathWorld, Wikipedia) are an extension of Bernoulli numbers. They obey this recurrence relation:

where is replaced by after the right-hand side has been expanded symbolically.

They can also be derived as coefficients in the following power series expansion:

The hopeful fact about Bernoulli polynomials in our case is that

,

which means that the equation whose solutions we are investigating,

boils down to

or even

.

Now is a polynomial of degree *n*+1 in *x*, so that in the area we’re looking in, where *x-n* is approximately 0.3*x*, will be infinitesimal in comparison with , so that the simplified equation

has the same asymptotic behaviour as the original.

Another hopeful line of inquiry is that is the coefficient of in the power series expansion of , while is the coefficient of in the expansion of .

Unfortunately the the *x* we are looking for is not constant but depends on *n*, so that we can’t establish interesting facts about and as a whole. All we can really do to isolate a particular power of *t* is to differentiate each of these two functions *n* times with respect to *t* and compare the results at *t*=0. Nevertheless, it is still good to find some sort of an occurrence of an exponential function, given that we are trying to get a reason for *ln *2 appearing in the result.