Let $V$ be the vector space polynomials of degree less than $n$ over $\mathbb{R}$

Question

Let $V$ be the vector space polynomials of degree less than $n$ over $\mathbb{R}$. In other words

$$V = \{p\in R[x] \mid \deg p < n\}.$$

Let $T : V \rightarrow V$ be the map $T =\frac{d}{dx}$. From elementary calculus, $T$ is linear

and $T^n=0$ for some positive integer $n$. For $t\in \mathbb{R}$, let $H_t : V \rightarrow V$ be the map $$H_t(p(x)) = p(x + t),$$ $p(x) \in V$.

We need to Show that $$e^{tT}=H_t.$$ So, should I start as follows? As, for any $p(x)\in \mathscr{C}^{\infty}$ and I should expand using Taylor expansion, $p(x+t)$ in the small neighborhood of t and which follows the desired result.

Any help will be highly appreciated.

Answer 1

I will do it when $n=4$; the general case is similar.

So, suppose that $p(x)=a_0+a_1x+a_2x^2+a_3x^3$. Then\begin{align}e^{tT}p(x)&=p(x)+tp'(x)+\frac{t^2}{2!}p''(x)+\frac{t^3}{3!}p'''(x)\\&=a_0+a_1x+a_2x^2+a_3x^2+t(a_1+2a_2x+3a_3x^2)+t^2(a_2+3a_3x)+t^3a_3\end{align}and, on the other hand,\begin{align}p(x+t)&=a_0+a_1(x+t)+a_2(x+t)^2+a_3(x+t)^3\\&=a_0+a_1x+a_2x^2+a_3x^2+t(a_1+2a_2x+3a_3x^2)+t^2(a_2+3a_3x)+t^3a_3.\end{align}