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

$f(s)=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n(\genfrac{}{}{0ex}{}{s}{n})a_n=\sum _{n=0}^{\mathrm{\infty }}\frac{(-s{)}_n}{n!}{a}_n$

where

$(\genfrac{}{}{0ex}{}{s}{n})$

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

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

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:

$\psi (s+1)=-\gamma -\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n}(\genfrac{}{}{0ex}{}{s}{n}).$

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

$\left\{\begin{array}{c}n\\ k\end{array}\right\}=\frac{1}{k!}\sum _{j=0}^k(-1{)}^{k-j}(\genfrac{}{}{0ex}{}{k}{j})j^n.$

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

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

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

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

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

The trigonometric functions have umbral identities:

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

and

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

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 xⁿ.

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

$\sum _{k=0}{B}_k{z}^k=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}\frac{tz}{e^{tz}-1}dt=\sum _{k=1}\frac{z}{(kz+1{)}^2}.$

The general relation gives the Newton series

$\sum _{k=0}\frac{B_k(x)}{z^k}\frac{(\genfrac{}{}{0ex}{}{1-s}{k})}{s-1}=z^{s-1}\zeta (s,x+z),$Template:Citation needed

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 $\frac{1}{\mathrm{\Gamma }(x)}=\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{x-a}{k})\sum _{j=0}^k\frac{(-1{)}^{k-j}}{\mathrm{\Gamma }(a+j)}(\genfrac{}{}{0ex}{}{k}{j}),$ 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)

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

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 integralsTemplate:Dead link", Theoretical Computer Science 144 (1995) pp 101–124.