In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence written in the form

where

is the binomial coefficient and is the rising factorial. Newtonian series often appear in relations of the form seen in umbral calculus.

## List[edit]

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 *k*th forward difference of the monomial *x*^{n} 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 *x*^{n}.

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)

## See also[edit]

## References[edit]

- Philippe Flajolet and Robert Sedgewick, "Mellin transforms and asymptotics: Finite differences and Rice's integrals
^{[permanent dead link]}",*Theoretical Computer Science**144*(1995) pp 101–124.