{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(f_n)_{n \\ge 0}$ be a sequence of complex numbers.\nThe *Poisson–Mellin–Newton cycle* of $(f_n)$ is the following chain of three objects.\n\n1. The *Poisson generating function* $g(t) = e^{-t} \\sum_{n \\ge 0} f_n \\frac{t^n}{n!}$, for $t \u003e 0$ where the series converges.\n2. The *Newton coefficients* $b_m = \\sum_{k=0}^{m} \\binom{m}{k} (-1)^k f_k$, which satisfy $g(t) = \\sum_{m \\ge 0} b_m \\frac{(-t)^m}{m!}$ and $f_n = \\sum_{k=0}^{n} \\binom{n}{k} (-1)^k b_k$.\n3. The [[Mellin transform|N5PVGRJE1T]] $\\varphi(s) = \\int_0^\\infty g(t)\\, t^{s-1}\\, dt$, where it converges.\n\nAssume that $b_m = b(m)$ for a function $b$ that is [[holomorphic|0NAHXR1FRH]] and of moderate growth in a half-plane.\nThen the cycle closes by the relations\n$$\\varphi(s) = \\Gamma(s)\\, b(-s), \\qquad f_n = \\frac{(-1)^n}{2\\pi i} \\oint_{C} b(s)\\, \\frac{n!}{s (s - 1) \\cdots (s - n)}\\, ds,$$\nwhere $C$ is a positively oriented contour that encircles the points $0, 1, \\dots, n$ and no singularity of $b$.\n\n## Remarks\n\nThe two identities in clause 2 come from multiplying the power series of $e^{-t}$ and $\\sum f_n t^n / n!$ and comparing coefficients.\nThe map $(f_n) \\mapsto (b_m)$ is the binomial transform, and it is its own inverse.\nThe first closing relation is an instance of [[Ramanujan's master theorem|ZN4NMX1DB4]], which needs growth conditions on $b$.\nThe second is the Nörlund–Rice integral: the residue of $n!\\,/\\,(s (s-1) \\cdots (s-n))$ at $s = k$ is $(-1)^{n-k} \\binom{n}{k}$, so the residue theorem gives back $f_n = \\sum_k \\binom{n}{k} (-1)^k b(k)$.\nHere $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\n\nExample: $f_n = (1/2)^n$.\nThen $g(t) = e^{-t/2}$, the Newton coefficients are $b_m = (1 - 1/2)^m = 2^{-m}$, so $b(s) = 2^{-s}$.\nIndeed $\\varphi(s) = 2^{s}\\, \\Gamma(s) = \\Gamma(s)\\, b(-s)$ for $\\operatorname{Re} s \u003e 0$.\n\nNon-example: for $f_n = (n!)^2$ the series $\\sum_n f_n t^n / n!$ diverges for every $t \u003e 0$, so there is no Poisson generating function and the cycle does not start.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "Y0PGRJHBYR",
    "kind": "definition",
    "msc": [
      "44A15",
      "39A70",
      "68W40"
    ],
    "name": "Poisson–Mellin–Newton cycle",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/N%C3%B8rlund%E2%80%93Rice_integral"
      }
    ],
    "tag": "AVSV6QBB6K",
    "uses": [
      "N5PVGRJE1T",
      "0H5SYBSED9",
      "ZN4NMX1DB4",
      "0NAHXR1FRH"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/AVSV6QBB6K/",
  "links": [
    {
      "field": "defines",
      "target": "Y0PGRJHBYR"
    },
    {
      "field": "uses",
      "target": "N5PVGRJE1T"
    },
    {
      "field": "uses",
      "target": "0H5SYBSED9"
    },
    {
      "field": "uses",
      "target": "ZN4NMX1DB4"
    },
    {
      "field": "uses",
      "target": "0NAHXR1FRH"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AVSV6QBB6K.md",
  "msc": [
    "44A15",
    "39A70",
    "68W40"
  ],
  "name": "Poisson–Mellin–Newton cycle",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AVS/AVSV6QBB6K.md",
  "status": null,
  "tag": "AVSV6QBB6K",
  "uri": "m2://AVSV6QBB6K",
  "url": "https://tamnd.github.io/math2/wiki/poisson-mellin-newton-cycle-definition/",
  "writing": {
    "tag": "AVSV6QBB6K",
    "kind": "definition",
    "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
      }
    ]
  }
}