Table of Newtonian series
# Table of Newtonian series

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

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

where

    ${s \choose n}$

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

## List

The generalized binomial theorem gives

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

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

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

The digamma function:

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

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

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

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

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

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

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

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

The trigonometric functions have umbral identities:

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

and

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

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

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

which can be recognized as resembling the Taylor series for sin x, with $(s)_n$ standing in the place of $x^n.$

In analytic number theory it is of interest to sum

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

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

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

The general relation gives the Newton series

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

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

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

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