{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$ and $0 \\le \\alpha \\le n$ be integers.\nLet $\\gamma$ be a positively oriented simple closed curve in $\\C$ that passes through none of the integers $0, \\dots, n$.\nAssume that its interior contains $\\alpha, \\alpha + 1, \\dots, n$ and none of $0, 1, \\dots, \\alpha - 1$.\nLet $f$ be holomorphic on an open set that contains $\\gamma$ and its interior.\nThen\n$$\\sum_{k=\\alpha}^{n} \\binom{n}{k} (-1)^k f(k) = \\frac{(-1)^n}{2\\pi i} \\oint_\\gamma f(z)\\, \\frac{n!}{z(z-1)(z-2)\\cdots(z-n)}\\, dz,$$\nwhere the integral is a [[contour integral|00FWNP9TXY]] and $\\binom{n}{k}$ is the [[binomial coefficient|092GEF38YD]].\n\n## Remarks\n\nThe proof is the [[residue theorem|032HKYS329]].\nInside $\\gamma$ the integrand has simple poles only at $z = k$ with $\\alpha \\le k \\le n$, and the residue there is $f(k)\\, n! / \\prod_{j \\ne k} (k - j) = (-1)^{n-k} \\binom{n}{k} f(k)$.\nMultiplying by $(-1)^n$ gives the terms $(-1)^k \\binom{n}{k} f(k)$.\nFor $\\alpha = 0$ the left side is $(-1)^n \\Delta^n f(0)$, an [[$n$-th forward difference|E38801KVJ1]], so the formula turns a high-order difference into a contour integral.\nExample: for $f = 1$ and $\\alpha = 0$, the left side is $(1 - 1)^n = 0$.\nOn the right, the integrand is $O(\\abs{z}^{-2})$, and the integral over a circle does not depend on its radius once $\\gamma$ encloses $0, \\dots, n$.\nSo it equals its limit as the radius grows, which is $0$.\n\n## Explanations\n\n### L1 human\n\nAlternating sums with binomial coefficients, such as $\\sum_k \\binom{n}{k} (-1)^k f(k)$, involve huge cancellations and are hard to estimate term by term.\nThe Nörlund–Rice integral rewrites such a sum as one complex integral.\nThe integral can then be estimated by moving the contour, which often gives precise asymptotics as $n$ grows.\n\n### L2 human\n\nThe integrand is $f(z)$ times the kernel $n!/(z(z-1)\\cdots(z-n)) = \\Gamma(n+1)\\Gamma(z-n)/\\Gamma(z+1)$, so its size away from the integers is controlled by the gamma function.\nTo get asymptotics one extends $f$ meromorphically, enlarges the contour, and collects the residues at the poles of $f$ that are crossed.\nThis method is a standard tool in the analysis of algorithms, for example for digital trees and related recursive structures.\nIt is closely related to the Mellin transform method for harmonic sums.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "3039GWNHWS"
    ],
    "kind": "statement",
    "msc": [
      "39A70",
      "30E20"
    ],
    "name": "Nörlund–Rice integral",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q3333616",
      "wikipedia": "Nørlund–Rice integral"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/N%C3%B8rlund%E2%80%93Rice_integral"
      }
    ],
    "tag": "PGMNNRN5YN",
    "uses": [
      "E38801KVJ1",
      "00FWNP9TXY",
      "032HKYS329",
      "092GEF38YD"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/PGMNNRN5YN/",
  "links": [
    {
      "field": "about",
      "target": "3039GWNHWS"
    },
    {
      "field": "uses",
      "target": "E38801KVJ1"
    },
    {
      "field": "uses",
      "target": "00FWNP9TXY"
    },
    {
      "field": "uses",
      "target": "032HKYS329"
    },
    {
      "field": "uses",
      "target": "092GEF38YD"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/PGMNNRN5YN.md",
  "msc": [
    "39A70",
    "30E20"
  ],
  "name": "Nörlund–Rice integral",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PGM/PGMNNRN5YN.md",
  "status": null,
  "tag": "PGMNNRN5YN",
  "uri": "m2://PGMNNRN5YN",
  "url": "https://tamnd.github.io/math2/wiki/norlund-rice-integral/",
  "writing": {
    "tag": "PGMNNRN5YN",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}