We set, for all $n \in \mathbb { N }$, $$D _ { n } ( X ) = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \therefore & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & C _ { n } & C _ { n + 1 } & \ldots & C _ { 2 n - 2 } & C _ { 2 n - 1 } \\ 1 & X & \cdots & X ^ { n - 2 } & X ^ { n - 1 } & X ^ { n } \end{array} \right|$$ and $$H _ { n } ^ { \prime } = \left| \begin{array} { c c c c c c } C _ { 1 } & C _ { 2 } & C _ { 3 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 2 } & \therefore & & \ddots & \ddots & C _ { n + 1 } \\ C _ { 3 } & & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 3 } \\ C _ { n - 1 } & \therefore & \ddots & & \ddots & C _ { 2 n - 2 } \\ C _ { n } & C _ { n + 1 } & \cdots & C _ { 2 n - 3 } & C _ { 2 n - 2 } & C _ { 2 n - 1 } \end{array} \right|.$$ Deduce that $\forall n \in \mathbb { N } , D _ { n } = U _ { n }$, then determine, for all $n \in \mathbb { N } ^ { * }$, the value of the determinant $H _ { n } ^ { \prime }$.
We set, for all $n \in \mathbb { N }$,
$$D _ { n } ( X ) = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \therefore & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & C _ { n } & C _ { n + 1 } & \ldots & C _ { 2 n - 2 } & C _ { 2 n - 1 } \\ 1 & X & \cdots & X ^ { n - 2 } & X ^ { n - 1 } & X ^ { n } \end{array} \right|$$
and
$$H _ { n } ^ { \prime } = \left| \begin{array} { c c c c c c } C _ { 1 } & C _ { 2 } & C _ { 3 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 2 } & \therefore & & \ddots & \ddots & C _ { n + 1 } \\ C _ { 3 } & & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & & C _ { 2 n - 3 } \\ C _ { n - 1 } & \therefore & \ddots & & \ddots & C _ { 2 n - 2 } \\ C _ { n } & C _ { n + 1 } & \cdots & C _ { 2 n - 3 } & C _ { 2 n - 2 } & C _ { 2 n - 1 } \end{array} \right|.$$
Deduce that $\forall n \in \mathbb { N } , D _ { n } = U _ { n }$, then determine, for all $n \in \mathbb { N } ^ { * }$, the value of the determinant $H _ { n } ^ { \prime }$.