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

:$f(s) = \sum_^\infty (-1)^n a_n = \sum_^\infty \frac a_n$

where

:$$

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

:$(1+z)^s = \sum_^z^n = 1+z+z^2+\cdots.$

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

: $(1+z) \frac = s (1+z)^s.$

The digamma function:

:$\psi(s+1)=-\gamma-\sum_^\infty \frac .$

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

:$\left\ =\frac\sum_^(-1)^ j^n.$

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

:$\Delta^k x^n = \sum_^(-1)^ (x+j)^n.$

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

:$\sum_^n \frac = \frac = \frac= B(n+1,s-n),s \notin \$

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

The trigonometric functions have umbral identities:

:$\sum_^\infty (-1)^n = 2^ \cos \frac$

and
:$\sum_^\infty (-1)^n = 2^ \sin \frac$

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

:$s - \frac + \frac - \frac + \cdots$

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
:$\!\sum_B_k z^k,$
where ''B'' are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as
:$\sum_B_k z^k= \int_0^\infty e^ \fracd t= \sum_\frac z.$
The general relation gives the Newton series
:$\sum_\frac\frac= z^\zeta(s,x+z),$
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
$\frac 1= \sum_^\infty \sum_^k \frac,$
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)
:$f(x)=\sum_ \sum_^k (-1)^f(a+j h).$

See also

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

References

* Philippe Flajolet and Robert Sedgewick,
Mellin transforms and asymptotics: Finite differences and Rice's integrals
, ''Theoretical Computer Science'' ''144'' (1995) pp 101–124.

{{Isaac Newton

Finite differences
Factorial and binomial topics
Newton series