Gustavo Colonnetti

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

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

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

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

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:

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

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 x^n.

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

$\sum _{k=0}\frac{B_k(x)}{z^k}\frac{(\genfrac{}{}{0ex}{}{1-s}{k})}{s-1}=z^{s-1}\zeta (s,x+z),$Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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 integrals", Theoretical Computer Science 144 (1995) pp 101–124.