Nørlund–Rice integral

In mathematics, the Nörlund-Rice integral relates a path integral on the complex plane to the sum of the residues of its poles, in such a way that the integrand involves the Pochhammer symbols or falling factorials, while the sum involves the binomial coefficients. As such, it commonly appears in the theory of finite differences. It is named after Niels Erik Nörlund and Rice.

Definition

The Nörlund-Rice integral is given by

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

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.

References

• Niels Erik Nörlund, Vorlesungen uber Differenzenrechnung, (1954) Chelsea Publishing Company, New York.