grandes-ecoles 2023 Q34

grandes-ecoles · France · centrale-maths1__official Sequences and Series Functional Equations and Identities via Series
By choosing $Q = E_1 - I$, prove that, if $p$ is a non-constant polynomial, then $$p'(X) = \sum_{k=1}^{\deg(p)} \frac{1}{k} \left( \sum_{j=0}^k (-1)^{j+1} \binom{k}{j} p(X+j) \right)$$ This is the formula for numerical differentiation of polynomials.
By choosing $Q = E_1 - I$, prove that, if $p$ is a non-constant polynomial, then
$$p'(X) = \sum_{k=1}^{\deg(p)} \frac{1}{k} \left( \sum_{j=0}^k (-1)^{j+1} \binom{k}{j} p(X+j) \right)$$
This is the formula for numerical differentiation of polynomials.