In mathematics, the Nørlund–Rice integral, sometimes called Rice's method, relates the nth forward difference of a function to a contour integral on the complex plane. It commonly appears in the theory of finite differences and has also 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
Δ n [ f ] ( x ) = ∑ k = 0 n ( n k ) ( − 1 ) n − k f ( x + k ) {\displaystyle \Delta ^{n}[f](x)=\sum _{k=0}^{n}{n \choose k}(-1)^{n-k}f(x+k)}
where ( n k ) {\displaystyle {n \choose k}} is the binomial coefficient. The Nørlund–Rice integral is given by
∑ k = α n ( n k ) ( − 1 ) n − k f ( k ) = n ! 2 π i ∮ γ f ( z ) z ( z − 1 ) ( z − 2 ) ⋯ ( z − n ) d z {\displaystyle \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)}}\,dz}
where f is understood to be meromorphic, α is an integer, 0 ≤ α ≤ n {\displaystyle 0\leq \alpha \leq n} , and the contour of integration is understood to circle the poles located at the integers α, ..., n, but encircles neither integers 0, ..., α − 1 {\displaystyle \alpha -1} nor any of the poles of f. The integral may also be written as
∑ k = α n ( n k ) ( − 1 ) k f ( k ) = − 1 2 π i ∮ γ B ( n + 1 , − z ) f ( z ) d z {\displaystyle \sum _{k=\alpha }^{n}{n \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 ) {\displaystyle 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
∑ k = α n ( n k ) ( − 1 ) n − k f ( k ) = − n ! 2 π i ∫ c − i ∞ c + i ∞ f ( z ) z ( z − 1 ) ( z − 2 ) ⋯ ( z − n ) d z {\displaystyle \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)}}\,dz}
where the constant c is to the left of α.
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