TITLE: Root of a special polynomial
QUESTION [5 upvotes]: Given a polynomial $P(x)=\sum_{n=0}^{d}a_nx^n\in\mathbb{R}[x]$ with all roots on the unit circle.
Question: Is it true that all the roots of $Q(x)=\sum_{n=0}^{d}a_n{{x+d-n}\choose{d}}$ lie on a straight line? If it is not true,is it possible to give a counterexample?

REPLY [2 votes]: With the help of Polya and Szego's book "Problems and Theorems in Analysis(II,Chapter 3,Problem 196.1)",I found this paper:
www.math.utexas.edu/~villegas/publications/hilbert.ps
It is just what I want.