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

:$f(s)\; =\; sum\_\{n=0\}^infty\; (-1)^n\; \{schoose\; n\}\; a\_n\; =\; sum\_\{n=0\}^infty\; frac\{(-s)\_n\}\{n!\}\; a\_n$

where

:$\{s\; choose\; k\}$

is the binomial coefficient and $(s)\_n$ is the rising 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

:

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

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

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

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

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 $x^n$.

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.