For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$ We define, for all $n \in \mathbb{N}^*$, the polynomial $S_n$ by: $S_n = a_0^{(n)} + a_1^{(n)} X + \cdots + a_{n-1}^{(n)} X^{n-1}$. The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the orthonormal family defined in question II.E. Express $s_n$ using the sequence of polynomials $\left(K_p\right)_{p \in \mathbb{N}}$.
For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is:
$$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$
We define, for all $n \in \mathbb{N}^*$, the polynomial $S_n$ by: $S_n = a_0^{(n)} + a_1^{(n)} X + \cdots + a_{n-1}^{(n)} X^{n-1}$. The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the orthonormal family defined in question II.E.
Express $s_n$ using the sequence of polynomials $\left(K_p\right)_{p \in \mathbb{N}}$.