Power series such that the $n$-th partial sum has $n$ distinct roots

Question

I am looking for examples of power series of the form

$$\sum_{k=0}^\infty a_k x^k$$

(where $a_k \in \mathbb{C}$ for all $k$) such that the polynomial given by its $n$-th partial sum has $n$ distinct roots, i.e.:

$$\sum_{k=0}^n a_k x^k$$

has $n$ distinct roots.

So far, I have found this family of examples: $\sum_{k=0}^\infty x^k$ and all of its derivatives. Can you help me find some examples that are not any of these?

Answer 1

I believe that the exponential function $\sum_{n=0}^{\infty} \dfrac{x^n}{n!} $ does this.

Here is one paper discussion the roots of the partial sums, although I don't see a proof that all the roots are distinct.

https://sites.math.washington.edu/~morrow/336_09/papers/Ian.pdf

After further search, I found this, which does prove that all the zeros are simple:

https://maa.tandfonline.com/doi/abs/10.1080/00029890.2005.11920265

It is the American Mathematical Monthly, 2005, vol. 10, p. 891.

Answer 2

Let $a_k$ be a sequence of numbers that are algebraically independent over the rationals. Thus for any nontrivial polynomial $p(x_0, \ldots, x_n)$ with rational coefficients, $p(a_0, \ldots, a_n) \ne 0$.

The polynomial $P_n(x) = a_0 + a_1 x + \ldots + a_n x^n$ has a repeated root if and only if its discriminant is $0$. In this case that discriminant is a nontrivial polynomial in $a_0, \ldots, a_n$ with integer coefficients, and therefore must be nonzero.