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" }, { "__type": "InlineCode", "__tag": 4051, "value": "r = n % k" }, { "__type": "Text", "__tag": 4046, "value": "), which can be put together into something valid for all integer values " }, { "__type": "InlineCode", "__tag": 4051, "value": "n >= 0" }, { "__type": "Text", "__tag": 4046, "value": " & " }, { "__type": "InlineCode", "__tag": 4051, "value": "k > 0" }, { "__type": "Text", "__tag": 4046, "value": " " } ] }, { "__type": "Code", "__tag": 4050, "value": "n!(k) = k ** ((n - r)/k) * gamma(n/k + 1) / gamma(r/k + 1) * max(r, 1)", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This is the basis of the approximation when " }, { "__type": "InlineCode", "__tag": 4051, "value": "exact=False" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In principle, any fixed choice of " }, { "__type": "InlineCode", "__tag": 4051, "value": "r" }, { "__type": "Text", "__tag": 4046, "value": " (ignoring its relation " }, { "__type": "InlineCode", "__tag": 4051, "value": "r = n%k" }, { "__type": "Text", "__tag": 4046, "value": " to " }, { "__type": "InlineCode", "__tag": 4051, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": ") would provide a suitable analytic continuation from integer " }, { "__type": "InlineCode", "__tag": 4051, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " to complex " }, { "__type": "InlineCode", "__tag": 4051, "value": "z" }, { "__type": "Text", "__tag": 4046, "value": " (not only satisfying the functional equation but also being logarithmically convex, c.f. 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For example," } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "factorialk(17, 4) = 17!!!! = 17 * 13 * 9 * 5 * 1" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In particular, for any integer " }, { "__type": "InlineCode", "__tag": 4051, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": ", we have" } ] }, { "__type": "Blockquote", "__tag": 4059, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "factorialk(n, 1) = factorial(n)" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "factorialk(n, 2) = factorial2(n)" } ] } ] } ], "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/_basic.py", "item_line": 3052, "item_type": "function", "aliases": [ "scipy.special.factorialk" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Code", "__tag": 4050, "value": "from scipy.special import factorialk\nfactorialk(5, k=1, exact=True)\nfactorialk(5, k=3, exact=True)\nfactorialk([5, 7, 9], k=3, exact=True)\nfactorialk([5, 7, 9], k=3, exact=False)\n", "execution_status": "success" } ], "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": "k", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "exact", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "extend", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "zero" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "factorialk" }, "references": [ ".. [1] Complex extension to multifactorial", " https://en.wikipedia.org/wiki/Double_factorial#Alternative_extension_of_the_multifactorial" ], "qa": "scipy.special._basic:factorialk", "arbitrary": [], "local_refs": [ "exact", "extend", "k", "n", "nf" ] }