We denote for $x \in \mathbb{R}_{+}$ and $m \in \mathbb{R}$ $$T_{m}(x) = \int_{0}^{\infty} t^{m} e^{-\left(t^{2}+x/t\right)} dt$$
a) Let $x \in \mathbb{R}_{+}^{*}$ and $m \in \mathbb{R}$. Perform the change of variable $t = 1/u$ in the integral defining $T_{m}$. Justify the calculation carefully.
b) Let $n \in \mathbb{N}$. Justify the existence of the quantity $\int_{0}^{\infty} u^{n} e^{-u} du$ and calculate it.
c) Show that for $m \in \mathbb{N} - \{0,1\}$, $T_{-m}(1) \leq (m-2)!$.
d) Let $k \in \mathbb{N}$. Show that the radius of convergence $R$ of the power series $\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} T_{k-n}(1) x^{n}$ satisfies $R \geq 1$.
e) Let $k \in \mathbb{N}$. Show that for $x \in ]-1,1[$, $$T_{k}(1+x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} T_{k-n}(1) x^{n}$$