The generalized binomial theorem gives

A proof for this identity can be obtained by showing that it satisfies the differential equation

The
of the gamma function, and its derivative the digamma function, can both have Newtonian series found by taking their binomial transform as sequences over the integers:

These are both valid in the right half-plane
, as proven by Charles Hermite in 1900[1] and Moritz Abraham Stern in 1847 (see Digamma function#Newton series) respectively.
The Stirling numbers of the second kind are given by the finite sum

This formula is a special case of the kth forward difference of the monomial xn evaluated at x = 0:

A related identity forms the basis of the Nörlund–Rice integral:

where
is the Gamma function and
is the Beta function.
The trigonometric functions have umbral identities:

and

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
[citation needed]
where
is the Hurwitz zeta function and
the Bernoulli polynomial. The series does not converge, the identity holds formally.
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)
