{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $P_n$ denote the [[Legendre polynomial|QFYGJVA996]] of degree $n \\ge 0$.\nThe *shifted Legendre polynomial* of degree $n$ is the polynomial\n$$\\tilde{P}_n(x) = P_n(2x - 1).$$\nEquivalently,\n$$\\tilde{P}_n(x) = (-1)^n \\sum_{k=0}^{n} \\binom{n}{k} \\binom{n+k}{k} (-x)^k = \\frac{1}{n!} \\frac{d^n}{dx^n} \\left( x^2 - x \\right)^n.$$\n\n## Remarks\n\nThe first ones are $\\tilde{P}_0 = 1$, $\\tilde{P}_1 = 2x - 1$ and $\\tilde{P}_2 = 6x^2 - 6x + 1$.\nCheck for $\\tilde{P}_2$: with $P_2(t) = (3t^2 - 1)/2$ and $t = 2x - 1$, one gets $(12x^2 - 12x + 2)/2 = 6x^2 - 6x + 1$.\nThey are orthogonal on $[0, 1]$: $\\int_0^1 \\tilde{P}_m(x) \\tilde{P}_n(x) \\, dx = \\frac{\\delta_{mn}}{2n + 1}$.\nFor instance $\\int_0^1 (2x - 1) \\, dx = 0$.\nNon-example: the unshifted polynomials are not orthogonal on $[0, 1]$, since $\\int_0^1 P_0(x) P_1(x) \\, dx = \\int_0^1 x \\, dx = \\frac12 \\ne 0$.\nSome authors, and some software libraries, use the reversed convention $P_n(1 - 2x) = (-1)^n \\tilde{P}_n(x)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "9V3QA0KTW8",
    "kind": "definition",
    "msc": [
      "33C45",
      "42C05"
    ],
    "name": "Shifted Legendre polynomials",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Legendre_polynomials"
      }
    ],
    "tag": "2B9G13MQK3",
    "uses": [
      "QFYGJVA996"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/2B9G13MQK3/",
  "links": [
    {
      "field": "defines",
      "target": "9V3QA0KTW8"
    },
    {
      "field": "uses",
      "target": "QFYGJVA996"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/2B9G13MQK3.md",
  "msc": [
    "33C45",
    "42C05"
  ],
  "name": "Shifted Legendre polynomials",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2B9/2B9G13MQK3.md",
  "status": null,
  "tag": "2B9G13MQK3",
  "uri": "m2://2B9G13MQK3",
  "url": "https://tamnd.github.io/math2/wiki/shifted-legendre-polynomials-definition/",
  "writing": {
    "tag": "2B9G13MQK3",
    "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
      }
    ]
  }
}