Some relations between the values of a function and its first derivative at $n$ abscissa points

Abstract:

For a polynomial, P, of degree $2n - 2$, there exists a relation between the values of P and the values of its first derivative, $P’$, at the n abscissa points ${x_1}, \cdots ,{x_n}$, \[ \sum \limits _{i = 1}^n {[{a_i}P({x_i}) + {b_i}P’({x_i})] = 0.} \] Replacing P by a differentiable function y yields \[ \sum \limits _{i = 1}^n {[{a_i}y({x_i}) + {b_i}y’({x_i})] = E(y,x).} \] These relations are obtained and the error function $E(y,x)$ is given explicitly.