Power series coefficients

Question

I've been trying for days now to find a closed form for the coefficients of the power series about $x=0$ of the function $$ f(x)=\exp\left(r^2\frac{x(n-2)-x^2(n-1)+x^n}{(x-1)^2}\right), $$ but I always end up with an infinite series at best. Can it be done at all?

Answer 1

Just an idea, but a bit too long for a comment. You have

$$\frac{f'(x)}{f(x)}=\frac{r^2}{(x-1)^3}\left((n-2)x^n-nx^{n-1}+nx-(n-2)\right)$$

So $f$ is a solution of the differential equation

$$(x-1)^3y'=r^2\left((n-2)x^n-nx^{n-1}+nx-(n-2)\right)y$$

Now you can try to plug the series

$$y=\sum_{k=0}^{\infty}a_kx^k$$

And then find a relation between coefficients. See for example on Wikipedia, Power series solution of differential equations