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| In [[mathematics]], a [[Newtonian series]], named after [[Isaac Newton]], is a sum over a [[sequence]] <math>a_n</math> written in the form
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| :<math>f(s) = \sum_{n=0}^\infty (-1)^n {s\choose n} a_n = \sum_{n=0}^\infty \frac{(-s)_n}{n!} a_n</math>
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| where
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| :<math>{s \choose k}</math>
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| is the [[binomial coefficient]] and <math>(s)_n</math> is the [[rising factorial]]. Newtonian series often appear in relations of the form seen in [[umbral calculus]].
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| ==List==
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| The generalized [[binomial theorem]] gives
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| :<math> (1+z)^{s} = \sum_{n = 0}^{\infty}{s \choose n}z^n =
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| 1+{s \choose 1}z+{s \choose 2}z^2+\cdots.</math>
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| A proof for this identity can be obtained by showing that it satisfies the differential equation
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| : <math> (1+z) \frac{d(1+z)^s}{dz} = s (1+z)^s.</math>
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| The [[digamma function]]:
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| :<math>\psi(s+1)=-\gamma-\sum_{n=1}^\infty \frac{(-1)^n}{n} {s \choose n}</math>
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| The [[Stirling numbers of the second kind]] are given by the finite sum
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| :<math>\left\{\begin{matrix} n \\ k \end{matrix}\right\}
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| =\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}{k \choose j} j^n.</math>
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| This formula is a special case of the ''k''th [[forward difference]] of the [[monomial]] ''x''<sup>''n''</sup> evaluated at ''x'' = 0:
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| :<math> \Delta^k x^n = \sum_{j=0}^{k}(-1)^{k-j}{k \choose j} (x+j)^n.</math>
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| A related identity forms the basis of the [[Nörlund–Rice integral]]:
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| :<math>\sum_{k=0}^n {n \choose k}\frac {(-1)^k}{s-k} =
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| \frac{n!}{s(s-1)(s-2)\cdots(s-n)} =
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| \frac{\Gamma(n+1)\Gamma(s-n)}{\Gamma(s+1)}=
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| B(n+1,s-n)</math>
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| where <math>\Gamma(x)</math> is the [[Gamma function]] and <math>B(x,y)</math> is the [[Beta function]].
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| The [[trigonometric function]]s have [[umbral calculus|umbral]] identities:
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| :<math>\sum_{n=0}^\infty (-1)^n {s \choose 2n} = 2^{s/2} \cos \frac{\pi s}{4}</math>
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| and
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| :<math>\sum_{n=0}^\infty (-1)^n {s \choose 2n+1} = 2^{s/2} \sin \frac{\pi s}{4}</math>
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| The umbral nature of these identities is a bit more clear by writing them in terms of the [[falling factorial]] <math>(s)_n</math>. The first few terms of the sin series are
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| :<math>s - \frac{(s)_3}{3!} + \frac{(s)_5}{5!} - \frac{(s)_7}{7!} + \cdots\,</math>
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| which can be recognized as resembling the [[Taylor series]] for sin ''x'', with (''s'')<sub>''n''</sub> standing in the place of ''x''<sup>''n''</sub>.
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| In [[analytic number theory]] it is of interest to sum
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| :<math>\!\sum_{k=0}B_k z^k,</math>
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| where ''B'' are the [[Bernoulli numbers]]. Employing the generating function its Borel sum can be evaluated as
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| :<math>\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>
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| The general relation gives the Newton series
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| :<math>\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|date=February 2012}}
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| where <math>\zeta</math> is the [[Hurwitz zeta function]] and <math>B_k(x)</math> the [[Bernoulli polynomials|Bernoulli polynomial]]. The series does not converge, the identity holds formally.
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| Another identity is
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| <math>\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>
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| which converges for <math>x>a</math>. This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)
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| :<math>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>
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| ==See also==
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| * [[Binomial transform]]
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| * [[List of factorial and binomial topics]]
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| * [[Nörlund–Rice integral]]
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| * [[Carlson's theorem]]
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| ==References==
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| * Philippe Flajolet and Robert Sedgewick, "[http://www-rocq.inria.fr/algo/flajolet/Publications/mellin-rice.ps.gz Mellin transforms and asymptotics: Finite differences and Rice's integrals]", ''Theoretical Computer Science'' ''144'' (1995) pp 101–124.
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| [[Category:Finite differences]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Mathematics-related lists|Newton series]]
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I would like to introduce myself to you, I am Jayson Simcox but I don't like when people use my full title. My wife and I live psychic reading in Mississippi and I adore every day living right here. Distributing production is how he tends to make a living. What me and my family members adore is to climb but I'm thinking on beginning some thing new.