In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence written in the form
is the binomial coefficient and is the rising factorial. Newtonian series often appear in relations of the form seen in umbral calculus.
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
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:
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
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)