Let X and Y be two random variables on some $(\Omega, \mathcal{F}, \mathbb{P}).$ We can assume that Y is bounded. It is clear that X and Y are independent iff:
$\mathbb{E}[e^{iuX}e^{ivY}]=\mathbb{E}[e^{iuX}]\;\mathbb{E}[e^{ivY}], \ \forall u,v \in \mathbb{R}.$
My question is, can we show that X and Y are independent iff
$\mathbb{E}[e^{iuX}p(Y)]=\mathbb{E}[e^{iuX}]\;\mathbb{E}[p(Y)]$ $\forall u \in \mathbb{R} $ and for all polynomial functions $p(\cdot)$ ?
Thank you!