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

${\displaystyle f(s)=\sum _{n=0}^{\infty }(-1)^{n}{s \choose n}a_{n}=\sum _{n=0}^{\infty }{\frac {(-s)_{n)){n!))a_{n))$

where

${\displaystyle {s \choose n))$

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.

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The generalized binomial theorem gives

${\displaystyle (1+z)^{s}=\sum _{n=0}^{\infty }{s \choose n}z^{n}=1+{s \choose 1}z+{s \choose 2}z^{2}+\cdots .}$

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

${\displaystyle (1+z){\frac {d(1+z)^{s)){dz))=s(1+z)^{s}.}$

The digamma function:

${\displaystyle \psi (s+1)=-\gamma -\sum _{n=1}^{\infty }{\frac {(-1)^{n)){n)){s \choose n}.}$

The Stirling numbers of the second kind are given by the finite sum

${\displaystyle \left\((\begin{matrix}n\\k\end{matrix))\right\}={\frac {1}{k!))\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}j^{n}.}$

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

${\displaystyle \Delta ^{k}x^{n}=\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}(x+j)^{n}.}$

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

${\displaystyle \sum _{k=0}^{n}{n \choose k}{\frac {(-1)^{n-k)){s-k))={\frac {n!}{s(s-1)(s-2)\cdots (s-n)))={\frac {\Gamma (n+1)\Gamma (s-n)}{\Gamma (s+1)))=B(n+1,s-n),s\notin \{0,\ldots ,n\))$

where ${\displaystyle \Gamma (x)}$ is the Gamma function and ${\displaystyle B(x,y)}$ is the Beta function.

The trigonometric functions have umbral identities:

${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n}=2^{s/2}\cos {\frac {\pi s}{4))}$

and

${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}{s \choose 2n+1}=2^{s/2}\sin {\frac {\pi s}{4))}$

The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial ${\displaystyle (s)_{n))$. The first few terms of the sin series are

${\displaystyle s-{\frac {(s)_{3)){3!))+{\frac {(s)_{5)){5!))-{\frac {(s)_{7)){7!))+\cdots }$

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

${\displaystyle \!\sum _{k=0}B_{k}z^{k},}$

where B are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as

${\displaystyle \sum _{k=0}B_{k}z^{k}=\int _{0}^{\infty }e^{-t}{\frac {tz}{e^{tz}-1))\,dt=\sum _{k=1}{\frac {z}{(kz+1)^{2))}.}$

The general relation gives the Newton series

${\displaystyle \sum _{k=0}{\frac {B_{k}(x)}{z^{k))}{\frac {1-s \choose k}{s-1))=z^{s-1}\zeta (s,x+z),}$^{[citation needed]}

where ${\displaystyle \zeta }$ is the Hurwitz zeta function and ${\displaystyle B_{k}(x)}$ the Bernoulli polynomial. The series does not converge, the identity holds formally.

Another identity is ${\displaystyle {\frac {1}{\Gamma (x)))=\sum _{k=0}^{\infty }{x-a \choose k}\sum _{j=0}^{k}{\frac {(-1)^{k-j)){\Gamma (a+j))){k \choose j},}$ which converges for ${\displaystyle x>a}$. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)

${\displaystyle f(x)=\sum _{k=0}((\frac {x-a}{h)) \choose k}\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}f(a+jh).}$