Let $I = [-1,1]$, $\alpha > 0$, $a_{k,n} = \frac{2k+1}{2n}$ for $k \in \llbracket 0, n-1 \rrbracket$, $R_n \in \mathbb{R}_{2n-1}[X]$ the interpolating polynomial of $f_\alpha(x) = \frac{1}{\alpha^2+x^2}$ at $\{\pm a_{k,n} \mid k \in \llbracket 0,n-1\rrbracket\}$, and $Q_n(X) = \lambda_n \prod_{k=0}^{n-1}(x^2 - a_{k,n}^2)$. Deduce that, for all $x \in [ - 1,1 ]$,
$$f _ { \alpha } ( x ) - R _ { n } ( x ) = \frac { ( - 1 ) ^ { n } } { x ^ { 2 } + \alpha ^ { 2 } } \prod _ { k = 0 } ^ { n - 1 } \frac { x ^ { 2 } - a _ { k , n } ^ { 2 } } { \alpha ^ { 2 } + a _ { k , n } ^ { 2 } }.$$