Table of Newtonian series

In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence ${\displaystyle a_n}$ written in the form

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

where

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

is the binomial coefficient and ${\displaystyle (s{)}_n}$ is the falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus.

The generalized binomial theorem gives

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

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

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

The ${\displaystyle \log }$ of the gamma function, and its derivative the digamma function, can both have Newtonian series found by taking their binomial transform as sequences over the integers:

${\displaystyle \begin{array}{rl}\log (\mathrm{\Gamma }(s+1))& ={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}(\genfrac{}{}{0ex}{}{s}{n}){\displaystyle \sum _{k=1}^n}(-1{)}^{n-k}(\genfrac{}{}{0ex}{}{n-1}{k-1})\log (k)\\ \psi (s+1)+\gamma =H_s& ={\displaystyle \sum _{n=1}^{\mathrm{\infty }}}(\genfrac{}{}{0ex}{}{s}{n})\frac{(-1{)}^{n-1}}{n}\end{array}}$

These are both valid in the right half-plane ${\displaystyle \mathrm{\Re }(s)>0}$, as proven by Charles Hermite in 1900^[1] and Moritz Abraham Stern in 1847 (see Digamma function) respectively.

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

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

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

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

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

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

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

The trigonometric functions have umbral identities:

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

and

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

The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial ${\displaystyle (s{)}_n}$. The first few terms of the sin series are

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

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

${\displaystyle \sum _{k=0}{B}_k{z}^k,}$

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

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

The general relation gives the Newton series

${\displaystyle \sum _{k=0}\frac{B_k(x)}{z^k}\frac{(\genfrac{}{}{0ex}{}{1-s}{k})}{s-1}=z^{s-1}\zeta (s,x+z),}$ where ${\displaystyle \zeta }$ is the Hurwitz zeta function and ${\displaystyle B_k(x)}$ the Bernoulli polynomial. The series does not converge, the identity holds formally.

Another identity is ${\displaystyle \frac{1}{\mathrm{\Gamma }(x)}={\displaystyle \sum _{k=0}^{\mathrm{\infty }}}(\genfrac{}{}{0ex}{}{x-a}{k}){\displaystyle \sum _{j=0}^k}\frac{(-1{)}^{k-j}}{\mathrm{\Gamma }(a+j)}(\genfrac{}{}{0ex}{}{k}{j}),}$ which converges for ${\displaystyle x>a}$. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)

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

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

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

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