Table of Newtonian series

In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence an written in the form

f(s)=∑n=0∞(−1)n(sn)an=∑n=0∞(−s)nn!an

where

(sn)

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=∑n=0∞(sn)zn=1+(s1)z+(s2)z2+⋯.

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

(1+z)d(1+z)sdz=s(1+z)s.

The 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:

log⁡(Γ(s+1))=∑n=1∞(sn)∑k=1n(−1)n−k(n−1k−1)log⁡(k)ψ(s+1)+γ=Hs=∑n=1∞(sn)(−1)n−1n

These are both valid in the right half-plane ℜ(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

{nk}=1k!∑j=0k(−1)k−j(kj)jn.

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

Δkxn=∑j=0k(−1)k−j(kj)(x+j)n.

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

∑k=0n(nk)(−1)n−ks−k=n!s(s−1)(s−2)⋯(s−n)=Γ(n+1)Γ(s−n)Γ(s+1)=B(n+1,s−n),s∉{0,…,n}

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

The trigonometric functions have umbral identities:

∑n=0∞(−1)n(s2n)=2s/2cos⁡πs4

and

∑n=0∞(−1)n(s2n+1)=2s/2sin⁡πs4

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−(s)33!+(s)55!−(s)77!+⋯

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

∑k=0Bkzk,

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

∑k=0Bkzk=∫0∞e−ttzetz−1dt=∑k=1z(kz+1)2.

The general relation gives the Newton series

∑k=0Bk(x)zk(1−sk)s−1=zs−1ζ(s,x+z), where ζ is the Hurwitz zeta function and Bk(x) the Bernoulli polynomial. The series does not converge, the identity holds formally.

Another identity is 1Γ(x)=∑k=0∞(x−ak)∑j=0k(−1)k−jΓ(a+j)(kj), 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)=∑k=0(x−ahk)∑j=0k(−1)k−j(kj)f(a+jh).

## See also

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

## References

1. ↑ Davis, Philip J. (1959). "Leonhard Euler's Integral: A Historical Profile of the Gamma Function: In Memoriam: Milton Abramowitz". The American Mathematical Monthly 66 (10): 849-869. doi:10.2307/2309786. https://www.jstor.org/stable/2309786.

* 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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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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