# Table of Newtonian series

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

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

where

${\displaystyle }$

is the binomial coefficient and ${\displaystyle (s)_}$ is the rising factorial. Newtonian series often appear in relations of the form seen in umbral calculus.

## List

The generalized binomial theorem gives

${\displaystyle (1+z)^=\sum _^{\infty }z^=1+z+z^+\cdots .}$

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

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

The digamma function:

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

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

${\displaystyle \left\{{\beginn\\k\end}\right\}={\frac }\sum _^(-1)^j^.}$

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

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

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

${\displaystyle \sum _^{\frac {(-1)^}}={\frac }={\frac {\Gamma (n+1)\Gamma (s-n)}{\Gamma (s+1)}}=B(n+1,s-n),s\notin \}$

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

The trigonometric functions have umbral identities:

${\displaystyle \sum _^{\infty }(-1)^=2^\cos {\frac {\pi s}}}$

and

${\displaystyle \sum _^{\infty }(-1)^=2^\sin {\frac {\pi s}}}$

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

${\displaystyle s-{\frac {(s)_}}+{\frac {(s)_}}-{\frac {(s)_}}+\cdots }$

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

${\displaystyle \!\sum _B_z^,}$

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

${\displaystyle \sum _B_z^=\int _^{\infty }e^{-t}{\frac -1}}dt=\sum _{\frac {(kz+1)^}}.}$

The general relation gives the Newton series

${\displaystyle \sum _{\frac (x)}}}{\frac }=z^\zeta (s,x+z),}$[citation needed]

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

Another identity is ${\displaystyle {\frac {\Gamma (x)}}=\sum _^{\infty }\sum _^{\frac {(-1)^}{\Gamma (a+j)}},}$ 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)

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