In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence ${\displaystyle a_{n}}$ written in the form

where

is the binomial coefficient and ${\displaystyle (s)_{n}}$ is the falling 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 ${\displaystyle \log }$ 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 ${\displaystyle \Re (s)>0}$, as proven by Charles Hermite in 1900 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