We consider the Hilbert polynomials $$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$ Demonstrate that, for all $x \in ] - 1,1 [$ and all $q \in \mathbb { N } ^ { * }$, we have $$( 1 - x ) ^ { - q / 2 } = \sum _ { p = 0 } ^ { + \infty } H _ { p } \left( \frac { q } { 2 } + p - 1 \right) x ^ { p }$$
We consider the Hilbert polynomials
$$\left\{ \begin{array} { l } H _ { 0 } ( X ) = 1 \\ H _ { n } ( X ) = \frac { 1 } { n ! } \prod _ { k = 0 } ^ { n - 1 } ( X - k ) = \frac { X ( X - 1 ) \cdots ( X - n + 1 ) } { n ! } \quad \text { for all } n \in \mathbb { N } ^ { * } \end{array} \right.$$
Demonstrate that, for all $x \in ] - 1,1 [$ and all $q \in \mathbb { N } ^ { * }$, we have
$$( 1 - x ) ^ { - q / 2 } = \sum _ { p = 0 } ^ { + \infty } H _ { p } \left( \frac { q } { 2 } + p - 1 \right) x ^ { p }$$