Higher order extensions of Melzak's formula - Gould H.W. (Utilitas Mathematica, 2007)
An explicit summation formula for the generalized Harmonic series s p = σk=0n 1/(y + k)p is given using Riordan's cycle indicator. We then show how the formula of Melzak f(x + y) = y(ny + n) σk=0n (-1) k(kn) f(x - k)/y + k where f(x) is a polynomial of degree ≤ n, may be generalized and is in fact a special case of an expansion from Lagrange's interpolation formula. In this we use a general formula of Hoppe for higher order derivatives of composite functions.