For $n \in \mathbb { Z }$, we denote by $\mathcal { E } _ { n }$ the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that $$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$$ For $\alpha \in \mathbb { R }$, let $\varphi _ { \alpha }$ be the function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ defined by $$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad \varphi _ { \alpha } ( t ) = t ^ { \alpha }$$ For all $n \in \mathbb { Z } ^ { * }$, determine the real numbers $\alpha$ such that $\varphi _ { \alpha }$ belongs to $\mathcal { E } _ { n }$.
For $n \in \mathbb { Z }$, we denote by $\mathcal { E } _ { n }$ the space of functions $f$ in $\mathcal { C } ^ { 2 } \left( \mathbb { R } _ { + } ^ { * } , \mathbb { C } \right)$ such that
$$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad t ^ { 2 } f ^ { \prime \prime } ( t ) + t f ^ { \prime } ( t ) - n ^ { 2 } f ( t ) = 0$$
For $\alpha \in \mathbb { R }$, let $\varphi _ { \alpha }$ be the function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ defined by
$$\forall t \in \mathbb { R } _ { + } ^ { * } , \quad \varphi _ { \alpha } ( t ) = t ^ { \alpha }$$
For all $n \in \mathbb { Z } ^ { * }$, determine the real numbers $\alpha$ such that $\varphi _ { \alpha }$ belongs to $\mathcal { E } _ { n }$.