{
  "__type": "IngestedDoc",
  "__tag": 4010,
  "_content": {
    "Notes": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "The polynomials "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "C_n^{(\\alpha)}"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " are orthogonal over "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "[-1,1]"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " with weight function "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "(1 - x^2)^{(\\alpha - 1/2)}"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "."
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Warns": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Raises": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Yields": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Methods": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Returns": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Parameters",
          "__tag": 4026,
          "children": [
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "C",
              "annotation": "orthopoly1d",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Gegenbauer polynomial."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Gegenbauer (ultraspherical) polynomial."
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Receives": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Warnings": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Attributes": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    },
    "Parameters": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Parameters",
          "__tag": 4026,
          "children": [
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "n",
              "annotation": "int",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Degree of the polynomial."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "alpha",
              "annotation": "float",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Parameter, must be greater than -0.5."
                    }
                  ]
                }
              ]
            },
            {
              "__type": "DocParam",
              "__tag": 4016,
              "name": "monic",
              "annotation": "bool, optional",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "If "
                    },
                    {
                      "__type": "InlineRole",
                      "__tag": 4003,
                      "value": "True",
                      "domain": null,
                      "role": null,
                      "inventory": null
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": ", scale the leading coefficient to be 1. Default is "
                    },
                    {
                      "__type": "InlineRole",
                      "__tag": 4003,
                      "value": "False",
                      "domain": null,
                      "role": null,
                      "inventory": null
                    },
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Extended Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Defined to be the solution of"
            }
          ]
        },
        {
          "__type": "Math",
          "__tag": 4058,
          "value": "(1 - x^2)\\frac{d^2}{dx^2}C_n^{(\\alpha)}\n  - (2\\alpha + 1)x\\frac{d}{dx}C_n^{(\\alpha)}\n  + n(n + 2\\alpha)C_n^{(\\alpha)} = 0"
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "for "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "\\alpha > -1/2"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "; "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "C_n^{(\\alpha)}"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " is a polynomial of degree "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "n"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "."
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Other Parameters": {
      "__type": "Section",
      "__tag": 4015,
      "children": [],
      "title": [],
      "level": 0,
      "target": null
    }
  },
  "_ordered_sections": [
    "Summary",
    "Extended Summary",
    "Parameters",
    "Attributes",
    "Methods",
    "Returns",
    "Yields",
    "Receives",
    "Other Parameters",
    "Raises",
    "Warns",
    "Warnings",
    "Notes"
  ],
  "item_file": "/scipy/special/_orthogonal.py",
  "item_line": 1568,
  "item_type": "function",
  "aliases": [
    "scipy.special.gegenbauer"
  ],
  "example_section_data": {
    "__type": "Section",
    "__tag": 4015,
    "children": [
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "import numpy as np\nfrom scipy import special\nimport matplotlib.pyplot as plt\n",
        "execution_status": "success"
      },
      {
        "__type": "Text",
        "__tag": 4046,
        "value": "\nWe can initialize a variable ``p`` as a Gegenbauer polynomial using the\n`gegenbauer` function and evaluate at a point ``x = 1``.\n\n"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "p = special.gegenbauer(3, 0.5, monic=False)\np\n",
        "execution_status": "success"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "p(1)\n",
        "execution_status": "failure"
      },
      {
        "__type": "Text",
        "__tag": 4046,
        "value": "\nTo evaluate ``p`` at various points ``x`` in the interval ``(-3, 3)``,\nsimply pass an array ``x`` to ``p`` as follows:\n\n"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "x = np.linspace(-3, 3, 400)\ny = p(x)\n",
        "execution_status": "success"
      },
      {
        "__type": "Text",
        "__tag": 4046,
        "value": "\nWe can then visualize ``x, y`` using `matplotlib.pyplot`.\n\n"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "fig, ax = plt.subplots()\n",
        "execution_status": "success"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "ax.plot(x, y)\nax.set_title(\"Gegenbauer (ultraspherical) polynomial of degree 3\")\nax.set_xlabel(\"x\")\nax.set_ylabel(\"G_3(x)\")\n",
        "execution_status": "failure"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "plt.show()\n",
        "execution_status": "success"
      },
      {
        "__type": "Figure",
        "__tag": 4024,
        "value": {
          "__type": "RefInfo",
          "__tag": 4000,
          "module": "scipy",
          "version": "1.17.1",
          "kind": "assets",
          "path": "fig-1b4fd1734d874e60.png"
        }
      }
    ],
    "title": [],
    "level": 0,
    "target": null
  },
  "see_also": [],
  "signature": {
    "__type": "SignatureNode",
    "__tag": 4029,
    "kind": "function",
    "parameters": [
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "n",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "alpha",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": {
          "__type": "Empty",
          "__tag": 4031
        }
      },
      {
        "__type": "SigParam",
        "__tag": 4030,
        "name": "monic",
        "annotation": {
          "__type": "Empty",
          "__tag": 4031
        },
        "kind": "POSITIONAL_OR_KEYWORD",
        "default": "False"
      }
    ],
    "return_annotation": {
      "__type": "Empty",
      "__tag": 4031
    },
    "target_name": "gegenbauer"
  },
  "references": null,
  "qa": "scipy.special._orthogonal:gegenbauer",
  "arbitrary": [],
  "local_refs": [
    "C",
    "alpha",
    "monic",
    "n"
  ]
}