{
  "__type": "IngestedDoc",
  "__tag": 4010,
  "_content": {
    "Notes": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "For fixed "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "\\alpha"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": ", the polynomials "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "L_n^{(\\alpha)}"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " are orthogonal over "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "[0, \\infty)"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": " with weight function "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "e^{-x}x^\\alpha"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "."
            }
          ]
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "The Laguerre polynomials are the special case where "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "\\alpha = 0"
            },
            {
              "__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": "L",
              "annotation": "orthopoly1d",
              "desc": [
                {
                  "__type": "Paragraph",
                  "__tag": 4045,
                  "children": [
                    {
                      "__type": "Text",
                      "__tag": 4046,
                      "value": "Generalized Laguerre polynomial."
                    }
                  ]
                }
              ]
            }
          ]
        }
      ],
      "title": [],
      "level": 0,
      "target": null
    },
    "Summary": {
      "__type": "Section",
      "__tag": 4015,
      "children": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Generalized (associated) Laguerre 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 -1."
                    }
                  ]
                }
              ]
            },
            {
              "__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": "x\\frac{d^2}{dx^2}L_n^{(\\alpha)}\n  + (\\alpha + 1 - x)\\frac{d}{dx}L_n^{(\\alpha)}\n  + nL_n^{(\\alpha)} = 0,"
        },
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "where "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "\\alpha > -1"
            },
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "; "
            },
            {
              "__type": "InlineMath",
              "__tag": 4057,
              "value": "L_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": 586,
  "item_type": "function",
  "aliases": [
    "scipy.special.genlaguerre"
  ],
  "example_section_data": {
    "__type": "Section",
    "__tag": 4015,
    "children": [
      {
        "__type": "Text",
        "__tag": 4046,
        "value": "The generalized Laguerre polynomials are closely related to the confluent\nhypergeometric function :math:`{}_1F_1`:\n\n    .. math::\n        L_n^{(\\alpha)} = \\binom{n + \\alpha}{n} {}_1F_1(-n, \\alpha +1, x)\n\nThis can be verified, for example,  for :math:`n = \\alpha = 3` over the\ninterval :math:`[-1, 1]`:\n\n"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "import numpy as np\nfrom scipy.special import binom\nfrom scipy.special import genlaguerre\nfrom scipy.special import hyp1f1\nx = np.arange(-1.0, 1.0, 0.01)\nnp.allclose(genlaguerre(3, 3)(x), binom(6, 3) * hyp1f1(-3, 4, x))\n",
        "execution_status": "success"
      },
      {
        "__type": "Text",
        "__tag": 4046,
        "value": "\nThis is the plot of the generalized Laguerre polynomials\n:math:`L_3^{(\\alpha)}` for some values of :math:`\\alpha`:\n\n"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "import matplotlib.pyplot as plt\nx = np.arange(-4.0, 12.0, 0.01)\nfig, ax = plt.subplots()\n",
        "execution_status": "success"
      },
      {
        "__type": "Code",
        "__tag": 4050,
        "value": "ax.set_ylim(-5.0, 10.0)\nax.set_title(r'Generalized Laguerre polynomials $L_3^{\\alpha}$')\nfor alpha in np.arange(0, 5):\n    ax.plot(x, genlaguerre(3, alpha)(x), label=rf'$L_3^{(alpha)}$')\nplt.legend(loc='best')\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-3a5fb6b2ff325dc4.png"
        }
      }
    ],
    "title": [],
    "level": 0,
    "target": null
  },
  "see_also": [
    {
      "__type": "SeeAlsoItem",
      "__tag": 4028,
      "name": {
        "__type": "CrossRef",
        "__tag": 4002,
        "value": "hyp1f1",
        "reference": {
          "__type": "LocalRef",
          "__tag": 4022,
          "kind": "module",
          "path": "scipy.special.cython_special:hyp1f1"
        },
        "kind": "module"
      },
      "descriptions": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "confluent hypergeometric function"
            }
          ]
        }
      ],
      "type": null
    },
    {
      "__type": "SeeAlsoItem",
      "__tag": 4028,
      "name": {
        "__type": "CrossRef",
        "__tag": 4002,
        "value": "laguerre",
        "reference": {
          "__type": "LocalRef",
          "__tag": 4022,
          "kind": "module",
          "path": "scipy.special._orthogonal:laguerre"
        },
        "kind": "module"
      },
      "descriptions": [
        {
          "__type": "Paragraph",
          "__tag": 4045,
          "children": [
            {
              "__type": "Text",
              "__tag": 4046,
              "value": "Laguerre polynomial."
            }
          ]
        }
      ],
      "type": null
    }
  ],
  "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": "genlaguerre"
  },
  "references": [
    ".. [AS] Milton Abramowitz and Irene A. Stegun, eds.",
    "    Handbook of Mathematical Functions with Formulas,",
    "    Graphs, and Mathematical Tables. New York: Dover, 1972."
  ],
  "qa": "scipy.special._orthogonal:genlaguerre",
  "arbitrary": [],
  "local_refs": [
    "L",
    "alpha",
    "monic",
    "n"
  ]
}