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 $m \in \mathbb{R}$. Calculate $T_{m}(x)$ in terms of $T_{m-2}(x)$ and $T_{m-3}(x)$. For this, you may consider the quantity $$\int_{A}^{B} t^{m-1}\left(2t - x/t^{2}\right) e^{-\left(t^{2}+x/t\right)} dt$$ for $0 < A < B$. b) Let $m \in \mathbb{R}$. Find a relation between $x T_{m}^{\prime\prime\prime}(x)$, $T_{m}^{\prime\prime}(x)$ and $T_{m}(x)$ for $x \in \mathbb{R}_{+}^{*}$.
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 $m \in \mathbb{R}$. Calculate $T_{m}(x)$ in terms of $T_{m-2}(x)$ and $T_{m-3}(x)$. For this, you may consider the quantity
$$\int_{A}^{B} t^{m-1}\left(2t - x/t^{2}\right) e^{-\left(t^{2}+x/t\right)} dt$$
for $0 < A < B$.
b) Let $m \in \mathbb{R}$. Find a relation between $x T_{m}^{\prime\prime\prime}(x)$, $T_{m}^{\prime\prime}(x)$ and $T_{m}(x)$ for $x \in \mathbb{R}_{+}^{*}$.