# 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

:$f\left(s\right) = sum_\left\{n=0\right\}^infty \left(-1\right)^n \left\{schoose n\right\} a_n = sum_\left\{n=0\right\}^infty frac\left\{\left(-s\right)_n\right\}\left\{n!\right\} a_n$

where

:$\left\{s choose k\right\}$

is the binomial coefficient and $\left(s\right)_n$ is the rising factorial. Newtonian series often appear in relations of the form seen in umbral calculus.

List

The generalized binomial theorem gives

:$\left(1+z\right)^\left\{s\right\} = sum_\left\{n = 0\right\}^\left\{infty\right\}\left\{s choose n\right\}z^n = 1+\left\{s choose 1\right\}z+\left\{s choose 2\right\}z^2+cdots.$

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

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

The digamma function:

:$psi\left(s+1\right)=-gamma-sum_\left\{n=1\right\}^infty frac\left\{\left(-1\right)^n\right\}\left\{n\right\} \left\{s choose n\right\}$

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:

:$Delta^k x^n = sum_\left\{j=1\right\}^\left\{k\right\}\left(-1\right)^\left\{k-j\right\}\left\{k choose j\right\} \left(x+j\right)^n.$

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

:$sum_\left\{k=0\right\}^n \left\{n choose k\right\}frac \left\{\left(-1\right)^k\right\}\left\{s-k\right\} = frac\left\{n!\right\}\left\{s\left(s-1\right)\left(s-2\right)cdots\left(s-n\right)\right\} = frac\left\{Gamma\left(n+1\right)Gamma\left(s-n\right)\right\}\left\{Gamma\left(s+1\right)\right\}= B\left(n+1,s-n\right)$

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

The trigonometric functions have umbral identities:

:$sum_\left\{n=0\right\}^infty \left(-1\right)^n \left\{s choose 2n\right\} = 2^\left\{s/2\right\} cos frac\left\{pi s\right\}\left\{4\right\}$

and :$sum_\left\{n=0\right\}^infty \left(-1\right)^n \left\{s choose 2n+1\right\} = 2^\left\{s/2\right\} sin frac\left\{pi s\right\}\left\{4\right\}$

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

:$s - frac\left\{\left(s\right)_3\right\}\left\{3!\right\} + frac\left\{\left(s\right)_5\right\}\left\{5!\right\} - frac\left\{\left(s\right)_7\right\}\left\{7!\right\} + cdots,$

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

ee also

* Binomial transform
* List of factorial and binomial topics

References

* Philippe Flajolet and Robert Sedgewick, " [http://www-rocq.inria.fr/algo/flajolet/Publications/mellin-rice.ps.gz Mellin transforms and asymptotics: Finite differences and Rice's integrals] ", "Theoretical Computer Science" "'144" (1995) pp 101-124.

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