I was studying the Riemann zeta function, and I stumbled on the polynomials given by the recursion formula:
$$P_{k+1} = x(1-x) P'_k+x(k+1)P_k\; \text{ and }\; P_0=1$$
Can someone help me find an explicit formula for them?
For context: If you differentiate $1 / (1-x)$ (harmonic series), multiply by $x$ and evaluate at $-1$, you get the negative of the Dirichlet eta function calculated at $-1$.
We can differentiate and multiply by $x$ several times to get other values of the eta function. Let's define $f_{k+1} = x f'_k$, $f_0=-1/(1-x)$. I thought about looking only at the numerator, defining $P_k = -(1 -x)^{k+1} f_k$. This way I got the said recursion formula.