{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $P_k^{(\\alpha, \\beta)}$ denote the [[Jacobi polynomials|TRD7DSF2KR]].\nLet $\\alpha, \\beta$ be real numbers with $\\beta \\ge 0$ and $\\alpha + \\beta \\ge -2$, and let $n \\ge 0$ be an integer.\nThen for every $x \\in [-1, 1]$,\n$$\\sum_{k=0}^{n} \\frac{P_k^{(\\alpha, \\beta)}(x)}{P_k^{(\\beta, \\alpha)}(1)} \\ge 0.$$\n\n## Remarks\n\nThe denominators are positive: $P_k^{(\\beta, \\alpha)}(1) = \\frac{(\\beta + 1)_k}{k!} \u003e 0$ for $\\beta \\ge 0$.\nFor $\\beta = 0$ all denominators equal $1$, and the inequality says $\\sum_{k=0}^{n} P_k^{(\\alpha, 0)}(x) \\ge 0$ on $[-1, 1]$.\nCheck for $n = 1$ and $\\alpha = \\beta = 0$: the sum is $1 + x$, which is nonnegative on $[-1, 1]$ and vanishes at $x = -1$.\nAskey and Gasper proved it in 1976.\nA short proof by Ekhad (1993) treats the case $\\beta = 0$ in its ${}_3F_2$ form.\nAn identity found with the Wilf–Zeilberger method writes this ${}_3F_2$ as a positive combination of other ${}_3F_2$ series.\nThese are nonnegative by Clausen's formula, which expresses them as squares of ${}_2F_1$ series.\n\n## Explanations\n\n### L1 human\n\nJacobi polynomials take both positive and negative values on $[-1, 1]$.\nThe inequality says that a certain weighted running sum of them never goes below zero.\nIt is surprising because the single terms change sign many times.\nIts fame comes from its use by Louis de Branges in his 1985 proof of the Bieberbach conjecture.\n\n### L2 human\n\nSums of the form $\\sum_k c_k P_k^{(\\alpha, \\beta)}(x)$ are partial sums of Jacobi expansions, and their positivity is a question of positive summability kernels.\nThe case $\\beta = 0$ can be rewritten as the positivity of a terminating ${}_3F_2$ hypergeometric series on $0 \\le t \u003c 1$.\nDe Branges needed this ${}_3F_2$ form for integer values of $\\alpha$ to finish his proof.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "TRD7DSF2KR"
    ],
    "kind": "statement",
    "msc": [
      "33C45",
      "26D15"
    ],
    "name": "Askey–Gasper inequality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q4807194",
      "wikipedia": "Askey–Gasper inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Askey–Gasper_inequality"
      }
    ],
    "tag": "VJ7AMZB988",
    "uses": [
      "TRD7DSF2KR"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/VJ7AMZB988/",
  "links": [
    {
      "field": "about",
      "target": "TRD7DSF2KR"
    },
    {
      "field": "uses",
      "target": "TRD7DSF2KR"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/VJ7AMZB988.md",
  "msc": [
    "33C45",
    "26D15"
  ],
  "name": "Askey–Gasper inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VJ7/VJ7AMZB988.md",
  "status": null,
  "tag": "VJ7AMZB988",
  "uri": "m2://VJ7AMZB988",
  "url": "https://tamnd.github.io/math2/wiki/askey-gasper-inequality/",
  "writing": {
    "tag": "VJ7AMZB988",
    "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
      }
    ]
  }
}