Higher order extensions of Melzak's formula

Abstract

An explicit summation formula for the generalized Harmonic series s _p = σ_k=0ⁿ 1/(y + k)^p is given using Riordan's cycle indicator. We then show how the formula of Melzak f(x + y) = y(_n^{y + n}) σ_k=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.