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 $F ( X ) := \dfrac { X ^ { q - 1 } } { 1 + X ^ { p } }$, and set $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$.
Deduce the partial fraction decomposition of $F ( X )$ in $\mathbf { R } ( X )$:
$$F ( X ) = \frac { 1 - ( - 1 ) ^ { p } } { 2 } \cdot \frac { ( - 1 ) ^ { q - 1 } } { p } \cdot \frac { 1 } { X + 1 } - \frac { 2 } { p } \sum _ { k = 0 } ^ { \lfloor p / 2 \rfloor - 1 } F _ { k } ( X )$$
where, for all $0 \leq k \leq \lfloor p / 2 \rfloor - 1$,
$$F _ { k } ( X ) := \frac { \cos \left( q \theta _ { k } \right) X - \cos \left( ( q - 1 ) \theta _ { k } \right) } { X ^ { 2 } - 2 \cos \left( \theta _ { k } \right) X + 1 }$$