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 $x \in ] - 1,1 \left[ , \sum _ { i = 0 } ^ { + \infty } \left( \sum _ { j = 0 } ^ { + \infty } \left| a _ { i , j } ( x ) \right| \right) = \exp \left( \frac { | x | } { \sqrt { 1 - | x | } } \right) - 1 \right.$.