{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "13CH1ZF88V"
    },
    {
      "field": "uses",
      "target": "4JVVDAQ9XX"
    },
    {
      "field": "uses",
      "target": "53MWMX2HWT"
    },
    {
      "field": "uses",
      "target": "J9367QFD3G"
    },
    {
      "field": "about",
      "target": "53MWMX2HWT"
    },
    {
      "field": "about",
      "target": "J9367QFD3G"
    },
    {
      "field": "specializes",
      "target": "QFYGJVA996"
    },
    {
      "field": "specializes",
      "target": "Z4NV7VK9T9"
    }
  ],
  "body": "\n## Summary\n\nThe classical orthogonal polynomials are the three families of Jacobi, Laguerre and [[Hermite polynomials|MGACY98E9H]].\nThey include special cases such as the Gegenbauer, Legendre and [[Chebyshev polynomials|M6EAR1JHPP]].\nEach family is orthogonal for a weight on an interval.\nThe weights are of beta type on $(-1, 1)$, of gamma type on $(0, \\infty)$, and Gaussian on $\\R$.\nThey are the only orthogonal polynomial families that are eigenfunctions of a second-order differential operator, and they have Rodrigues formulas and explicit recurrences.\nThey are the basic tools of approximation, Gaussian quadrature, and separation of variables in mathematical physics.\n\n## Explanations\n\n### L1 human\n\nTwo polynomials are orthogonal for a weight when the integral of their product times the weight is zero, like perpendicular vectors.\nFor a given weight, the Gram–Schmidt process applied to $1, x, x^2, \\dots$ gives one orthogonal polynomial of each degree.\nFor three kinds of weight this produces families with especially nice formulas, and these are the classical ones.\nFor example, the weight $e^{-x^2}$ on the whole line gives the Hermite polynomials $1$, $2x$, $4x^2 - 2$, and so on.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "33C45",
      "42C05"
    ],
    "name": "Classical orthogonal polynomials",
    "same_as": {
      "wikidata": "Q17006917",
      "wikipedia": "Classical orthogonal polynomials"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Classical_orthogonal_polynomials"
      }
    ],
    "tag": "47WJ34NC9K"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/47WJ34NC9K/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/47WJ34NC9K.md",
  "msc": [
    "33C45",
    "42C05"
  ],
  "name": "Classical orthogonal polynomials",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/47W/47WJ34NC9K.md",
  "status": null,
  "tag": "47WJ34NC9K",
  "uri": "m2://47WJ34NC9K",
  "url": "https://tamnd.github.io/math2/wiki/classical-orthogonal-polynomials/",
  "writing": {
    "tag": "47WJ34NC9K",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}