We set, for all $n \in \mathbb { N }$, $$D _ { n } ( X ) = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \therefore & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & C _ { n } & C _ { n + 1 } & \ldots & C _ { 2 n - 2 } & C _ { 2 n - 1 } \\ 1 & X & \cdots & X ^ { n - 2 } & X ^ { n - 1 } & X ^ { n } \end{array} \right|$$ with the inner product $( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x$. Let $( n , k ) \in \mathbb { N } ^ { 2 }$ such that $k < n$. Show $\left( D _ { n } \mid X ^ { k } \right) = 0$.
We set, for all $n \in \mathbb { N }$,
$$D _ { n } ( X ) = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \therefore & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & C _ { n } & C _ { n + 1 } & \ldots & C _ { 2 n - 2 } & C _ { 2 n - 1 } \\ 1 & X & \cdots & X ^ { n - 2 } & X ^ { n - 1 } & X ^ { n } \end{array} \right|$$
with the inner product $( P \mid Q ) = \frac { 2 } { \pi } \int _ { 0 } ^ { 1 } P ( 4 x ) Q ( 4 x ) \frac { \sqrt { 1 - x } } { \sqrt { x } } \mathrm { ~d} x$. Let $( n , k ) \in \mathbb { N } ^ { 2 }$ such that $k < n$. Show $\left( D _ { n } \mid X ^ { k } \right) = 0$.