Table of Newtonian series

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

[math]\displaystyle{ f(s) = \sum_{n=0}^\infty (-1)^n {s\choose n} a_n = \sum_{n=0}^\infty \frac{(-s)_n}{n!} a_n }[/math]

where

[math]\displaystyle{ {s \choose n} }[/math]

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

## List

The generalized binomial theorem gives

[math]\displaystyle{ (1+z)^s = \sum_{n = 0}^{\infty}{s \choose n}z^n = 1+{s \choose 1}z+{s \choose 2}z^2+\cdots. }[/math]

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

[math]\displaystyle{ (1+z) \frac{d(1+z)^s}{dz} = s (1+z)^s. }[/math]

The digamma function:

[math]\displaystyle{ \psi(s+1)=-\gamma-\sum_{n=1}^\infty \frac{(-1)^n}{n} {s \choose n}. }[/math]

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

[math]\displaystyle{ \left\\{\begin{matrix} n \\\ k \end{matrix}\right\\} =\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}{k \choose j} j^n. }[/math]

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

[math]\displaystyle{ \Delta^k x^n = \sum_{j=0}^{k}(-1)^{k-j}{k \choose j} (x+j)^n. }[/math]

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

[math]\displaystyle{ \sum_{k=0}^n {n \choose k}\frac {(-1)^{n-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),s \notin \\{0,\ldots,n\\} }[/math]

where [math]\displaystyle{ \Gamma(x) }[/math] is the Gamma function and [math]\displaystyle{ B(x,y) }[/math] is the Beta function.

The trigonometric functions have umbral identities:

[math]\displaystyle{ \sum_{n=0}^\infty (-1)^n {s \choose 2n} = 2^{s/2} \cos \frac{\pi s}{4} }[/math]

and

[math]\displaystyle{ \sum_{n=0}^\infty (-1)^n {s \choose 2n+1} = 2^{s/2} \sin \frac{\pi s}{4} }[/math]

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

[math]\displaystyle{ s - \frac{(s)_3}{3!} + \frac{(s)_5}{5!} - \frac{(s)_7}{7!} + \cdots }[/math]

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

[math]\displaystyle{ \\!\sum_{k=0}B_k z^k, }[/math]

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

[math]\displaystyle{ \sum_{k=0}B_k z^k= \int_0^\infty e^{-t} \frac{t z}{e^{t z}-1}d t= \sum_{k=1}\frac z{(k z+1)^2}. }[/math]

The general relation gives the Newton series

[math]\displaystyle{ \sum_{k=0}\frac{B_k(x)}{z^k}\frac{{1-s\choose k}}{s-1}= z^{s-1}\zeta(s,x+z), }[/math][citation needed]

where [math]\displaystyle{ \zeta }[/math] is the Hurwitz zeta function and [math]\displaystyle{ B_k(x) }[/math] the Bernoulli polynomial. The series does not converge, the identity holds formally.

Another identity is [math]\displaystyle{ \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}, }[/math] which converges for [math]\displaystyle{ x\gt a }[/math]. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)

[math]\displaystyle{ f(x)=\sum_{k=0}{\frac{x-a}h \choose k} \sum_{j=0}^k (-1)^{k-j}{k\choose j}f(a+j h). }[/math]

## 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[no|permanent dead link|dead link}}]", Theoretical Computer Science 144 (1995) pp 101–124.

* v
* t
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Sir Isaac Newton

Publications|

* Fluxions (1671)
* De Motu (1684)
* Principia (1687; writing)
* Opticks (1704)
* Queries (1704)
* Arithmetica (1707)
* De Analysi (1711)

Other writings|

* Quaestiones (1661–65)
* "standing on the shoulders of giants" (1675)
* Notes on the Jewish Temple (c. 1680)
* "General Scholium" (1713; "hypotheses non fingo" )
* Ancient Kingdoms Amended (1728)
* Corruptions of Scripture (1754)

Contributions|

* Calculus
* fluxion
* Impact depth
* Inertia
* Newton disc
* Newton polygon
* Newton–Okounkov body
* Newton's reflector
* Newtonian telescope
* Newton scale
* Newton's metal
* Newton's cradle
* Spectrum
* Structural coloration

Newtonianism|

* Bucket argument
* Newton's inequalities
* Newton's law of cooling
* Newton's law of universal gravitation
* post-Newtonian expansion
* parameterized
* gravitational constant
* Newton–Cartan theory
* Schrödinger–Newton equation
* Newton's laws of motion
* Kepler's laws
* Newtonian dynamics
* Newton's method in optimization
* Apollonius's problem
* truncated Newton method
* Gauss–Newton algorithm
* 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
* Newton's notation
* Rotating spheres
* Newton's cannonball
* Newton–Cotes formulas
* Newton's method
* generalized Gauss–Newton method
* Newton fractal
* Newton's identities
* Newton polynomial
* Newton's theorem of revolving orbits
* Newton–Euler equations
* Newton number
* kissing number problem
* Newton's quotient
* Parallelogram of force
* Newton–Puiseux theorem
* Absolute space and time
* Luminiferous aether
* Newtonian series
* table

Personal life|

* Woolsthorpe Manor (birthplace)
* Cranbury Park (home)
* Early life
* Later life
* Religious views
* Occult studies
* Scientific Revolution
* Copernican Revolution

Relations|

* Catherine Barton (niece)
* John Conduitt (nephew-in-law)
* Isaac Barrow (professor)
* William Clarke (mentor)
* Benjamin Pulleyn (tutor)
* John Keill (disciple)
* William Stukeley (friend)
* William Jones (friend)
* Abraham de Moivre (friend)

Depictions|

* Newton by Blake (monotype)
* Newton by Paolozzi (sculpture)

Namesake|

* Isaac Newton Institute
* Isaac Newton Medal
* Isaac Newton Telescope
* Isaac Newton Group of Telescopes
* Newton (unit)

Categories|

Category Isaac Newton not found

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