For all $n \in \mathbb { N }$, set $\mu _ { n } = \left( X ^ { n } \mid 1 \right)$ with the inner product
$$( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x .$$
Deduce that $\forall n \in \mathbb { N } , \mu _ { n } = C _ { n }$.