We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$. We define the rational fraction $F ( X ) := \dfrac { X ^ { q - 1 } } { 1 + X ^ { p } }$. Show that there exist constants $\left( a _ { 0 } , b _ { 0 } , \ldots , b _ { \lfloor p / 2 \rfloor - 1 } \right) \in \mathbf { C } ^ { \lfloor p / 2 \rfloor + 1 }$ such that $$F ( X ) = \frac { 1 - ( - 1 ) ^ { p } } { 2 } \cdot \frac { a _ { 0 } } { X + 1 } + \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } \left( \frac { b _ { k } } { X - \omega _ { p , k } } + \frac { \overline { b _ { k } } } { X - \overline { \omega _ { p , k } } } \right) ,$$ where the $\omega _ { p , k }$ are constants which one will specify and $F ( X )$ is the rational fraction defined at the beginning of this part. In the case where $p$ is even, we set $a _ { 0 } = 0$.
We fix a pair $( p , q ) \in E _ { 3 } := \left\{ ( p , q ) \in \left( \mathbf { N } ^ { * } \right) ^ { 2 } : p > q \right\}$. We define the rational fraction $F ( X ) := \dfrac { X ^ { q - 1 } } { 1 + X ^ { p } }$.
Show that there exist constants $\left( a _ { 0 } , b _ { 0 } , \ldots , b _ { \lfloor p / 2 \rfloor - 1 } \right) \in \mathbf { C } ^ { \lfloor p / 2 \rfloor + 1 }$ such that
$$F ( X ) = \frac { 1 - ( - 1 ) ^ { p } } { 2 } \cdot \frac { a _ { 0 } } { X + 1 } + \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } \left( \frac { b _ { k } } { X - \omega _ { p , k } } + \frac { \overline { b _ { k } } } { X - \overline { \omega _ { p , k } } } \right) ,$$
where the $\omega _ { p , k }$ are constants which one will specify and $F ( X )$ is the rational fraction defined at the beginning of this part.
In the case where $p$ is even, we set $a _ { 0 } = 0$.