In mathematics, the Nörlund-Rice integral, sometimes called Rice's method, relates the nth 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 occurs in computer science and specifically graph theory as an estimate of binary tree lengths. It is named in honour of Niels Erik Nörlund and Stephen Oswald Rice. Nörlund's contribution was to define the integral; Rice's contribution was to demonstrate its utility by applying saddle-point techniques to its evaluation.

Definition

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The nth forward difference of a function f(x) is given by

$\backslash Delta^n*(x)=\; \backslash sum\_\{k=0\}^n\; \{n\; \backslash choose\; k\}\; (-1)^\{n-k\}\; f(x+k)$

where $\{n\; \backslash choose\; k\}$ is the binomial coefficient.

The Nörlund-Rice integral is given by

$\backslash sum\_\{k=\backslash alpha\}^n\; \{n\; \backslash choose\; k\}\; (-1)^\{n-k\}\; f(k)\; =$

\frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{z(z-1)(z-2)\cdots(z-n)}\, dz

where f is understood to be meromorphic, α is an integer, $0\backslash leq\; \backslash alpha\; \backslash leq\; n$, and the contour of integration is understood to circle the poles located at the integers α, ..., n, but none of the poles of f. The integral may also be written as

$\backslash sum\_\{k=\backslash alpha\}^n\; \{n\; \backslash choose\; k\}\; (-1)^\{k\}\; f(k)\; =$

-\frac{1}{2\pi i} \oint_\gamma B(n+1, -z) f(z)\, dz

where B(a,b) is the Euler beta function. If the function $f(z)$ 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

$\backslash sum\_\{k=\backslash alpha\}^n\; \{n\; \backslash choose\; k\}\; (-1)^\{n-k\}\; f(k)\; =$

\frac{-n!}{2\pi i} \int_{c-i\infty}^{c+i\infty} \frac{f(z)}{z(z-1)(z-2)\cdots(z-n)}\, dz

where the constant c is to the left of α.

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 Nörlund-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 $\backslash \{f\_n\backslash \}$ be a sequence, and let g(t) be the corresponding Poisson generating function, that is, let

$g(t)\; =\; e^\{-t\}\; \backslash sum\_\{n=0\}^\backslash infty\; f\_n\; t^n$

Taking its Mellin transform

$\backslash phi(s)=\backslash int\_0^\backslash infty\; g(t)\; t^\{s-1\}\; dt$

one can then regain the original sequence by means of the Nörlund-Rice integral:

$f\_n\; =\; \backslash frac\{(-1)^n\; \}\{2\backslash pi\; i\}$

\int_\gamma \frac {\phi(s)}{\Gamma(-s)} \frac{n!}{s(s-1)\cdots (s-n)} ds

where Γ is the gamma function.

Riesz mean

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A closely related integral frequently occurs in the discussion of Riesz means. 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.

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 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.

See also

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• Binomial transform
• Table of Newtonian series
• List of factorial and binomial topics

References

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• Niels Erik Nörlund, Vorlesungen uber Differenzenrechnung, (1954) Chelsea Publishing Company, New York.
• Donald E. Knuth, The Art of Computer Programming,(1973), Vol. 3 Addison-Wesley.
• Philippe Flajolet and Robert Sedgewick, "Mellin transforms and asymptotics: Finite differences and Rice's integrals", Theoretical Computer Science 144 (1995) pp 101-124.
• Peter Kirschenhofer, "A Note on Alternating Sums", The Electronic Journal of Combinatorics, Volume 3 (1996) Issue 2 Article 7.

Factorial and binomial topics | Complex analysis | Integral transforms | Finite differences | Graph theory