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 $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$. Calculate $a _ { 0 }$ in the case where $p$ is odd, then show that, for all integers $k \in \llbracket 0 , \lfloor p / 2 \rfloor - 1 \rrbracket$, $b _ { k }$ can be written in the form $$b _ { k } = - \frac { 1 } { p } e ^ { i q \theta _ { k } }$$
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 $\theta _ { k } := ( 2 k + 1 ) \dfrac { \pi } { p }$.
Calculate $a _ { 0 }$ in the case where $p$ is odd, then show that, for all integers $k \in \llbracket 0 , \lfloor p / 2 \rfloor - 1 \rrbracket$, $b _ { k }$ can be written in the form
$$b _ { k } = - \frac { 1 } { p } e ^ { i q \theta _ { k } }$$