Table of Newtonian series
The generalized binomial theorem gives
A proof for this identity can be obtained by showing that it satisfies the differential equation
The digamma function:
The Stirling numbers of the second kind are given by the finite sum
A related identity forms the basis of the Nörlund–Rice integral:
The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial . The first few terms of the sin series are
which can be recognized as resembling the Taylor series for sin x, with (s)n standing in the place of xn.
In analytic number theory it is of interest to sum
where B are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as
The general relation gives the Newton series
Another identity is which converges for . This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)
- Philippe Flajolet and Robert Sedgewick, "Mellin transforms and asymptotics: Finite differences and Rice's integrals", Theoretical Computer Science 144 (1995) pp 101–124.