Nörlund–Rice integral

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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 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 saddle-point techniques to its evaluation.

Definition

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

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

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

The Nörlund–Rice integral is given by

$\sum _{{k=\alpha }}^{n}{n \choose k}(-1)^{{n-k}}f(k)={\frac {n!}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z(z-1)(z-2)\cdots (z-n)}}\,{\mathrm {d}}z$

where f is understood to be meromorphic, α is an integer, $0\leq \alpha \leq n$ , 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

$\sum _{{k=\alpha }}^{n}{n \choose k}(-1)^{{k}}f(k)=-{\frac {1}{2\pi i}}\oint _{\gamma }B(n+1,-z)f(z)\,{\mathrm {d}}z$

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

$\sum _{{k=\alpha }}^{n}{n \choose k}(-1)^{{n-k}}f(k)={\frac {-n!}{2\pi i}}\int _{{c-i\infty }}^{{c+i\infty }}{\frac {f(z)}{z(z-1)(z-2)\cdots (z-n)}}\,{\mathrm {d}}z$

where the constant c is to the left of α.

Poisson–Mellin–Newton cycle

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 $\{f_{n}\}$ be a sequence, and let g(t) be the corresponding Poisson generating function, that is, let

$g(t)=e^{{-t}}\sum _{{n=0}}^{\infty }f_{n}t^{n}.$

Taking its Mellin transform

$\phi (s)=\int _{0}^{\infty }g(t)t^{{s-1}}\,{\mathrm {d}}t,$

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

$f_{n}={\frac {(-1)^{n}}{2\pi i}}\int _{\gamma }{\frac {\phi (s)}{\Gamma (-s)}}{\frac {n!}{s(s-1)\cdots (s-n)}}\,{\mathrm {d}}s$

where Γ is the gamma function.

Riesz mean

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

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.

References

Niels Erik Nørlund, 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.

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