Table of Newtonian series

The generalized binomial theorem gives

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

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

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

The digamma function:

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

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

:$\backslash left\backslash \{\backslash begin\{matrix\}\; n\; \backslash \backslash \; k\; \backslash end\{matrix\}\backslash right\backslash \}$

=\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:

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

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

:$\backslash sum\_\{k=0\}^n\; \{n\; \backslash choose\; k\}\backslash frac\; \{(-1)^\{n-k\}\}\{s-k\}\; =$

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

The trigonometric functions have umbral identities:

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

and

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

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

:$s\; -\; \backslash frac\{(s)\_3\}\{3!\}\; +\; \backslash frac\{(s)\_5\}\{5!\}\; -\; \backslash frac\{(s)\_7\}\{7!\}\; +\; \backslash cdots$

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

:$\backslash !\backslash sum\_\{k=0\}B\_k\; z^k,$

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

:$\backslash sum\_\{k=0\}B\_k\; z^k=\; \backslash int\_0^\backslash infty\; e^\{-t\}\; \backslash frac\{t\; z\}\{e^\{t\; z\}-1\}\; \backslash ,\; dt=\; \backslash sum\_\{k=1\}\backslash frac\; z\{(k\; z+1)^2\}.$

The general relation gives the Newton series

:$\backslash sum\_\{k=0\}\backslash frac\{B\_k(x)\}\{z^k\}\backslash frac\{\{1-s\backslash choose\; k\}\}\{s-1\}=\; z^\{s-1\}\backslash zeta(s,x+z),${{Citation needed|date=February 2012}}

where $\backslash zeta$ is the Hurwitz zeta function and $B\_k(x)$ the Bernoulli polynomial. The series does not converge, the identity holds formally.

Another identity is

$\backslash frac\; 1\{\backslash Gamma(x)\}=\; \backslash sum\_\{k=0\}^\backslash infty\; \{x-a\backslash choose\; k\}\backslash sum\_\{j=0\}^k\; \backslash frac\{(-1)^\{k-j\}\}\{\backslash Gamma(a+j)\}\{k\backslash choose\; j\},$

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)

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

• Binomial transform
• List of factorial and binomial topics
• Nörlund–Rice integral
• Carlson's theorem

• Philippe Flajolet and Robert Sedgewick, "[https://algo.inria.fr/flajolet/Publications/FlSe95.pdf Mellin transforms and asymptotics: Finite differences and Rice's integrals]", Theoretical Computer Science 144 (1995) pp 101–124.

{{Isaac Newton}}

Category:Finite differences

Category:Factorial and binomial topics

Newton series