The family $\left(K_p\right)_{p \in \mathbb{N}}$ is the unique family of polynomials such that for all $p \in \mathbb{N}$, the degree of $K_p$ equals $p$ with strictly positive leading coefficient, and for all $N \in \mathbb{N}$, $\left(K_p\right)_{0 \leqslant p \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for $\langle \cdot, \cdot \rangle$.
For all $p \in \llbracket 0; n-1 \rrbracket$, calculate $K_p(1)$.