Lucas's theorem and an application

by @sequoia · difficulty 65/100

Analysis

Status: Unreviewed

Let $P$ be a complex polynomial with only simple roots. Show that all the roots of $P'$ belong to the convex hull of the roots of $P$ , meaning the set $\left\{\sum_{i=1}^d \lambda_i z_i/ \forall i\in[\![1,d]\!] \lambda_i\geqslant0, \sum_{i=1}^d\lambda_i=1\right\}.$
Let $d\in\mathbb{N}^*$ and $u_0,\dots,u_{d-1}$ be some complex numbers and $(u_n)_{n\in\mathbb{N}}$ be the complex sequence defined by those first terms and the relation $\forall n\in\mathbb{N}, u_{n+d}=\frac{u_{n+d-1}+\dots+u_{n+1}+u_n}{d}.$ Show that $u$ converges to a complex $l$ that you shall specify.

Proofs

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