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 Subfactorial

 http://functions.wolfram.com/06.42.06.0002.01

 Input Form

 Subfactorial[z] \[Proportional] Subfactorial[Subscript[z, 0]] + (1/E) ((-I) Pi Gamma[1 + Subscript[z, 0], 0, -1] - (-1)^Subscript[z, 0] Gamma[1 + Subscript[z, 0]]^2 HypergeometricPFQRegularized[{1 + Subscript[z, 0], 1 + Subscript[z, 0]}, {2 + Subscript[z, 0], 2 + Subscript[z, 0]}, 1] + Gamma[1 + Subscript[z, 0]] PolyGamma[1 + Subscript[z, 0]]) (z - Subscript[z, 0]) + (1/(2 E)) ((-Pi^2) Gamma[1 + Subscript[z, 0], -1] + ((2 (-1)^Subscript[z, 0])/(1 + Subscript[z, 0])^3) ((-I) Pi (1 + Subscript[z, 0]) HypergeometricPFQ[{1 + Subscript[z, 0], 1 + Subscript[z, 0]}, {2 + Subscript[z, 0], 2 + Subscript[z, 0]}, 1] + HypergeometricPFQ[{1 + Subscript[z, 0], 1 + Subscript[z, 0], 1 + Subscript[z, 0]}, {2 + Subscript[z, 0], 2 + Subscript[z, 0], 2 + Subscript[z, 0]}, 1]) + Gamma[1 + Subscript[z, 0]] (Pi^2 + PolyGamma[1 + Subscript[z, 0]]^2 + PolyGamma[1, 1 + Subscript[z, 0]])) (z - Subscript[z, 0])^2 + O[(z - Subscript[z, 0])^3]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Subfactorial", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["Subfactorial", "[", SubscriptBox["z", "0"], "]"]], "+", RowBox[List[FractionBox["1", "\[ExponentialE]"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["z", "0"]]], ",", "0", ",", RowBox[List["-", "1"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["z", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["z", "0"]]], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["z", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["z", "0"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["z", "0"]]], ",", RowBox[List["2", "+", SubscriptBox["z", "0"]]]]], "}"]], ",", "1"]], "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["z", "0"]]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", SubscriptBox["z", "0"]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", "\[ExponentialE]"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[Pi]", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["z", "0"]]], ",", RowBox[List["-", "1"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["z", "0"]], " "]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]], "3"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["z", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["z", "0"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["z", "0"]]], ",", RowBox[List["2", "+", SubscriptBox["z", "0"]]]]], "}"]], ",", "1"]], "]"]]]], "+", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["z", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["z", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["z", "0"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["z", "0"]]], ",", RowBox[List["2", "+", SubscriptBox["z", "0"]]], ",", RowBox[List["2", "+", SubscriptBox["z", "0"]]]]], "}"]], ",", "1"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["z", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", SubscriptBox["z", "0"]]], "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", SubscriptBox["z", "0"]]]]], "]"]]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], " ", "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "3"], "]"]]]]]]]]

 MathML Form

 Subfactorial ( z ) Subfactorial ( z 0 ) + 1 ( - ( - 1 ) z 0 2 F ~ 2 ( z 0 + 1 , z 0 + 1 ; z 0 + 2 , z 0 + 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] Γ ( z 0 + 1 ) 2 + ψ TagBox["\[Psi]", PolyGamma] ( z 0 + 1 ) Γ ( z 0 + 1 ) - π Γ ( z 0 + 1 , 0 , - 1 ) ) ( z - z 0 ) + 1 2 ( - π 2 Γ ( z 0 + 1 , - 1 ) + Γ ( z 0 + 1 ) ( ψ TagBox["\[Psi]", PolyGamma] ( z 0 + 1 ) 2 + π 2 + ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( z 0 + 1 ) ) + 1 ( z 0 + 1 ) 3 ( ( 3 F 3 ( z 0 + 1 , z 0 + 1 , z 0 + 1 ; z 0 + 2 , z 0 + 2 , z 0 + 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox["F", "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] - π ( z 0 + 1 ) 2 F 2 ( z 0 + 1 , z 0 + 1 ; z 0 + 2 , z 0 + 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[SubscriptBox["z", "0"], "+", "2"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] ) ( 2 ( - 1 ) z 0 ) ) ) ( z - z 0 ) 2 + O ( ( z - z 0 ) 3 ) Proportional Subfactorial z Subfactorial Subscript z 0 1 -1 -1 -1 Subscript z 0 HypergeometricPFQRegularized Subscript z 0 1 Subscript z 0 1 Subscript z 0 2 Subscript z 0 2 1 Gamma Subscript z 0 1 2 PolyGamma Subscript z 0 1 Gamma Subscript z 0 1 -1 Gamma Subscript z 0 1 0 -1 z -1 Subscript z 0 1 2 -1 -1 2 Gamma Subscript z 0 1 -1 Gamma Subscript z 0 1 PolyGamma Subscript z 0 1 2 2 PolyGamma 1 Subscript z 0 1 1 Subscript z 0 1 3 -1 HypergeometricPFQ Subscript z 0 1 Subscript z 0 1 Subscript z 0 1 Subscript z 0 2 Subscript z 0 2 Subscript z 0 2 1 -1 Subscript z 0 1 HypergeometricPFQ Subscript z 0 1 Subscript z 0 1 Subscript z 0 2 Subscript z 0 2 1 2 -1 Subscript z 0 z -1 Subscript z 0 2 O z -1 Subscript z 0 3 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Subfactorial", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Subfactorial", "[", SubscriptBox["zz", "0"], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ",", "0", ",", RowBox[List["-", "1"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["zz", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["zz", "0"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["zz", "0"]]], ",", RowBox[List["2", "+", SubscriptBox["zz", "0"]]]]], "}"]], ",", "1"]], "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "\[ExponentialE]"], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[Pi]", "2"]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ",", RowBox[List["-", "1"]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["zz", "0"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["zz", "0"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["zz", "0"]]], ",", RowBox[List["2", "+", SubscriptBox["zz", "0"]]]]], "}"]], ",", "1"]], "]"]]]], "+", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["zz", "0"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["zz", "0"]]], ",", RowBox[List["2", "+", SubscriptBox["zz", "0"]]], ",", RowBox[List["2", "+", SubscriptBox["zz", "0"]]]]], "}"]], ",", "1"]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], ")"]], "3"]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", SubscriptBox["zz", "0"]]], "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", SubscriptBox["zz", "0"]]]]], "]"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], RowBox[List["2", " ", "\[ExponentialE]"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], "3"]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02