# Expand power of a polynomial

I'm very new to Mathematica, so excuse my innocence. I have the following expression:

$$\left( \sum_{n=0}^r \frac{(-1)^n}{n!} y^n \right)^f$$

I would like Mathematica to expand out the expression in powers of $y$, as a polynomial, where $r$ and $f$ are arbitrary positive integers. Is this possible?

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• Take the result from a given r, f (here called result), then Series[result, {y, Infinity, 0}] // Normal – ciao May 22 '16 at 7:41
• That will give an expression in terms of specific $r$,$f$. I desire an expression for general $r$ and $f$...In other words ungiven. – Anand Chotai May 22 '16 at 8:10

A bit complicated, this one:

With[{m = 5, r = 3},
CoefficientList[Sum[(-x)^n/n!, {n, 0, m}]^r, x] ==
Table[Sum[FactorialPower[r, k]
BellY[n, k, Table[(-1)^i, {i, m}]],
{k, 0, r}]/n!, {n, 0, m r}]]
True


Recall that the partial Bell polynomials are a way to express Faà di Bruno's formula, which applies here since the coefficients of a polynomial are the same as the successive derivatives of a polynomial divided by an appropriate factorial. There may be a less cumbersome closed form; I'll keep trying to look.

• Thanks a lot for that! A very good idea, I will examine it more closely. – Anand Chotai May 23 '16 at 4:03