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}}^{\infty }(-1)^{n}{s \choose n}a_{n}=\sum _{{n=0}}^{\infty }{\frac {(-s)_{n}}{n!}}a_{n}

where

{s \choose n}

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}}^{{\infty }}{s \choose n}z^{n}=1+{s \choose 1}z+{s \choose 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}}^{\infty }{\frac {(-1)^{n}}{n}}{s \choose n}.

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

\left\{{\begin{matrix}n\\k\end{matrix}}\right\}={\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:

\Delta^k x^n = \sum_{j=0}^{k}(-1)^{k-j}{k \choose j} (x+j)^n.

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

\sum _{{k=0}}^{n}{n \choose k}{\frac {(-1)^{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)

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

The trigonometric functions have umbral identities:

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

and

\sum _{{n=0}}^{\infty }(-1)^{n}{s \choose 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}=\int _{0}^{\infty }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 {{1-s \choose k}}{s-1}}=z^{{s-1}}\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 ${\frac 1{\Gamma (x)}}=\sum _{{k=0}}^{\infty }{x-a \choose k}\sum _{{j=0}}^{k}{\frac {(-1)^{{k-j}}}{\Gamma (a+j)}}{k \choose j},$ 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)=\sum _{k=0}{{\frac {x-a}{h}} \choose k}\sum _{j=0}^{k}(-1)^{k-j}{k \choose j}f(a+jh).

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

Publications	De analysi per aequationes numero terminorum infinitas (1669, published 1711) Method of Fluxions (1671) De motu corporum in gyrum (1684) Philosophiæ Naturalis Principia Mathematica (1687) General Scholium (1713) Opticks (1704) The Queries (1704) Arithmetica Universalis (1707)

Other writings	Notes on the Jewish Temple Quaestiones quaedam philosophicae The Chronology of Ancient Kingdoms Amended (1728) An Historical Account of Two Notable Corruptions of Scripture (1754)

Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation Post-Newtonian expansion Parameterized post-Newtonian formalism Newton–Cartan theory Schrödinger–Newton equation Gravitational constant Newton's laws of motion Newtonian dynamics Newton's method in optimization Gauss–Newton algorithm Truncated Newton method Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Newtonian fluid Corpuscular theory of light Leibniz–Newton calculus controversy Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method Newton fractal Generalized Gauss–Newton method Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number Kissing number problem Power number Solar mass Dynamics Absolute time and space Finite difference Table of Newtonian series Impact depth Structural coloration Inertia

Life	Cranbury Park Woolsthorpe Manor Early life of Isaac Newton Later life of Isaac Newton Religious views of Isaac Newton Isaac Newton's occult studies The Mysteryes of Nature and Art Scientific revolution Copernican Revolution

Friends and family	Catherine Barton John Conduitt William Clarke Benjamin Pulleyn William Stukeley William Jones Isaac Barrow Abraham de Moivre John Keill

Discoveries and inventions	Calculus Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Newton's cradle Sextant

Phrases	"Hypotheses non fingo" "Standing on the shoulders of giants"

Theory expansions	Kepler's laws of planetary motion Problem of Apollonius

Related	Writing of Principia Mathematica Newton (Blake) List of things named after Isaac Newton Newton (unit) Isaac Newton in popular culture Elements of the Philosophy of Newton Isaac Newton S/O Philipose

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Table of Newtonian series

List

See also

References