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| In [[mathematics]], the '''Nörlund–Rice integral''', sometimes called '''Rice's method''', relates the ''n''th [[forward difference]] of a function to a [[line integral]] on the [[complex plane]]. As such, it commonly appears in the theory of [[finite differences]], and also has been applied in [[computer science]] and [[graph theory]] to estimate [[binary tree]] lengths. It is named in honour of [[Niels Erik Nørlund]] and [[Stephen O. Rice]]. Nørlund's contribution was to define the integral; Rice's contribution was to demonstrate its utility by applying [[Method of steepest descent|saddle-point technique]]s to its evaluation.
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| ==Definition==
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| The ''n''th [[forward difference]] of a function ''f''(''x'') is given by
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| :<math>\Delta^n[f](x)= \sum_{k=0}^n {n \choose k} (-1)^{n-k} f(x+k)</math>
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| where <math>{n \choose k}</math> is the [[binomial coefficient]].
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| The Nörlund–Rice integral is given by
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| :<math>\sum_{k=\alpha}^n {n \choose k} (-1)^{n-k} f(k) =
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| \frac{n!}{2\pi i}
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| \oint_\gamma \frac{f(z)}{z(z-1)(z-2)\cdots(z-n)}\, \mathrm{d}z</math>
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| where ''f'' is understood to be [[meromorphic]], α is an integer, <math>0\leq \alpha \leq n</math>, and the contour of integration is understood to circle the [[pole (complex analysis)|poles]] located at the integers α, ..., ''n'', but none of the poles of ''f''. The integral may also be written as
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| :<math>\sum_{k=\alpha}^n {n \choose k} (-1)^{k} f(k) =
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| -\frac{1}{2\pi i}
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| \oint_\gamma B(n+1, -z) f(z)\, \mathrm{d}z</math>
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| where ''B''(''a'',''b'') is the Euler [[beta function]]. If the function <math>f(z)</math> is [[polynomially bounded]] on the right hand side of the complex plane, then the contour may be extended to infinity on the right hand side, allowing the transform to be written as
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| :<math>\sum_{k=\alpha}^n {n \choose k} (-1)^{n-k} f(k) =
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| \frac{-n!}{2\pi i}
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| \int_{c-i\infty}^{c+i\infty} \frac{f(z)}{z(z-1)(z-2)\cdots(z-n)}\, \mathrm{d}z</math>
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| where the constant ''c'' is to the left of α.
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| ==Poisson–Mellin–Newton cycle==
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| The Poisson–Mellin–Newton cycle, noted by Flajolet et al. in 1985, is the observation that the resemblance of the Nørlund–Rice integral to the [[Mellin transform]] is not accidental, but is related by means of the [[binomial transform]] and the [[Newton series]]. In this cycle, let <math>\{f_n\}</math> be a [[sequence]], and let ''g''(''t'') be the corresponding [[Poisson generating function]], that is, let
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| :<math>g(t) = e^{-t} \sum_{n=0}^\infty f_n t^n.</math>
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| Taking its Mellin transform
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| :<math>\phi(s)=\int_0^\infty g(t) t^{s-1}\, \mathrm{d}t,</math>
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| one can then regain the original sequence by means of the Nörlund–Rice integral:
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| :<math>f_n = \frac{(-1)^n }{2\pi i}
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| \int_\gamma
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| \frac {\phi(s)}{\Gamma(-s)} \frac{n!}{s(s-1)\cdots (s-n)}\, \mathrm{d}s</math>
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| where Γ is the [[gamma function]].
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| ==Riesz mean==
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| A closely related integral frequently occurs in the discussion of [[Riesz mean]]s. Very roughly, it can be said to be related to the Nörlund–Rice integral in the same way that [[Perron's formula]] is related to the Mellin transform: rather than dealing with infinite series, it deals with finite series.
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| ==Utility==
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| The integral representation for these types of series is interesting because the integral can often be evaluated using [[asymptotic expansion]] or [[Method of steepest descent|saddle-point]] techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large ''n''.
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| ==See also==
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| * [[Table of Newtonian series]]
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| * [[List of factorial and binomial topics]]
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| ==References==
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| * Niels Erik Nørlund, ''Vorlesungen uber Differenzenrechnung'', (1954) Chelsea Publishing Company, New York.
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| * Donald E. Knuth, ''The Art of Computer Programming'', (1973), Vol. 3 Addison-Wesley.
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| * Philippe Flajolet and Robert Sedgewick, "[http://www-rocq.inria.fr/algo/flajolet/Publications/FlSe95.pdf Mellin transforms and asymptotics: Finite differences and Rice's integrals]", ''Theoretical Computer Science'' '''144''' (1995) pp 101–124.
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| * Peter Kirschenhofer, "[http://www.combinatorics.org/Volume_3/volume3_2.html#R7 A Note on Alternating Sums]", ''[http://www.combinatorics.org The Electronic Journal of Combinatorics]'', Volume '''3''' (1996) Issue 2 Article 7.
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| {{DEFAULTSORT:Norlund-Rice integral}}
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| [[Category:Factorial and binomial topics]]
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| [[Category:Complex analysis]]
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| [[Category:Integral transforms]]
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| [[Category:Finite differences]]
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Greetings! I am Myrtle Shroyer. The favorite hobby for my children and me is to perform baseball and I'm trying to make it a occupation. Since she was 18 she's been working as a receptionist but her promotion by no means comes. For a while I've been in South Dakota and my mothers and fathers reside nearby.
My webpage - at home std test