{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "QVCKQ46TFM"
    }
  ],
  "body": "\n## Summary\n\nThe continuous $q$-Hermite polynomials $H_n(x \\mid q)$ are a family of orthogonal polynomials on $[-1, 1]$ depending on a base $q$.\nThey are $q$-analogues of the [[Hermite polynomials|MGACY98E9H]], to which they tend after rescaling as $q \\to 1$.\nAt $q = 0$ they are the Chebyshev polynomials of the second kind.\nThey are one of the simplest families of the $q$-Askey scheme, a special case of the [[Rogers polynomials|7H7SACRYEF]] and of the [[Askey–Wilson polynomials|RSNZNG0WA1]].\nThey appear in $q$-deformed probability and in the theory of $q$-Gaussian processes.\n\n## Explanations\n\n### L1 human\n\nWrite $x = \\cos\\theta$ and form the sum of $e^{i(n - 2k)\\theta}$ over $k = 0, \\dots, n$, each weighted by a [[Gaussian binomial coefficient|YS09Z1YNEK]] $\\binom{n}{k}_q$.\nThe result is a polynomial in $x$ of degree $n$, the continuous $q$-Hermite polynomial.\nAt $q = 0$ all weights are $1$, and the sum is $\\frac{\\sin((n+1)\\theta)}{\\sin\\theta}$.\nFor $n = 2$ one gets $H_2(x \\mid q) = 4x^2 - (1 - q)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "33D45"
    ],
    "name": "Continuous q-Hermite polynomials",
    "same_as": {
      "wikidata": "Q5165508",
      "wikipedia": "Continuous q-Hermite polynomials"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Continuous_q-Hermite_polynomials"
      }
    ],
    "tag": "XFKVZJ55Q5"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/XFKVZJ55Q5/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/XFKVZJ55Q5.md",
  "msc": [
    "33D45"
  ],
  "name": "Continuous q-Hermite polynomials",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XFK/XFKVZJ55Q5.md",
  "status": null,
  "tag": "XFKVZJ55Q5",
  "uri": "m2://XFKVZJ55Q5",
  "url": "https://tamnd.github.io/math2/wiki/continuous-q-hermite-polynomials/",
  "writing": {
    "tag": "XFKVZJ55Q5",
    "kind": "concept",
    "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
      }
    ]
  }
}