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 Hypergeometric0F1

 http://functions.wolfram.com/07.17.26.0145.01

 Input Form

 Hypergeometric0F1[b, z] BesselK[b + n, 2 Sqrt[z]] == Pi (-1)^(n - 1) 2^(-(3/2) + b) Gamma[b] (Sqrt[Pi] Csc[(((-1)^n + 4 b)/4) Pi] MeijerG[{{(1 - b)/2, 1 - b/2}, {(1 + 2 b)/4, (3 - 2 b)/4}}, {{1 - (b - n)/2, (b + n)/2}, {(1 + 2 b)/4, (3 - 2 b)/4, -((b + n)/2), 1 - (3 b + n)/2}}, 4 z] - ((Csc[b Pi] 2^(1/2 - b - n))/(z^((b + n)/2) Sqrt[Pi])) Sum[(((-1)^Floor[(1 + n)/2] 4^k z^k Gamma[1/2 + k - n + Floor[n/2]])/ (k! Gamma[b + k] Gamma[1 - b + k - n])) Pochhammer[1 - k + Floor[n/2], n - Floor[n/2]], {k, 0, Floor[n/2]}]) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Hypergeometric0F1", "[", RowBox[List["b", ",", "z"]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["b", "+", "n"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]], "\[Equal]", RowBox[List["\[Pi]", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", "b"]]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["\[Pi]"], RowBox[List["Csc", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "+", RowBox[List["4", "b"]]]], "4"], " ", "\[Pi]"]], "]"]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "b"]], "2"], ",", RowBox[List["1", "-", FractionBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], "4"], " ", ",", FractionBox[RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], "4"]]], " ", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["b", "-", "n"]], "2"]]], ",", FractionBox[RowBox[List["b", "+", "n"]], "2"]]], " ", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], "4"], " ", ",", FractionBox[RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], "4"], " ", ",", RowBox[List["-", FractionBox[RowBox[List["b", "+", "n"]], "2"]]], ",", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["3", " ", "b"]], "+", "n"]], "2"]]]]], " ", "}"]]]], "}"]], ",", RowBox[List["4", "z"]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "b", "-", "n"]]], " ", SuperscriptBox["z", RowBox[List["-", FractionBox[RowBox[List["b", "+", "n"]], "2"]]]], " "]], SqrtBox["\[Pi]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "n"]], "2"], "]"]]], " ", SuperscriptBox["4", "k"], SuperscriptBox["z", "k"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "b", "+", "k", "-", "n"]], "]"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 0 F 1 ( ; b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric0F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", Hypergeometric0F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric0F1] K b + n ( 2 z ) π ( - 1 ) n - 1 2 b - 3 2 Γ ( b ) ( π csc ( 1 4 ( 4 b + ( - 1 ) n ) π ) G 4 , 6 2 , 2 ( 4 z 1 - b 2 , 1 - b 2 , 1 4 ( 2 b + 1 ) , 1 4 ( 3 - 2 b ) 1 - b - n 2 , b + n 2 , 1 4 ( 2 b + 1 ) , 1 4 ( 3 - 2 b ) , - 1 2 ( b + n ) , 1 - 1 2 ( 3 b + n ) ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "6"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["4", " ", "z"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "b"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["b", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox[RowBox[List["b", "-", "n"]], "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["b", "+", "n"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["b", "+", "n"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "b"]], "+", "n"]], ")"]]]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] - csc ( b π ) 2 - b - n + 1 2 z - b + n 2 π k = 0 n 2 ( - 1 ) n + 1 2 4 k z k Γ ( k - n + n 2 + 1 2 ) ( 1 - k + n 2 ) n - n 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "k", "+", RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]]], ")"]], RowBox[List["n", "-", RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]]]], Pochhammer] k ! Γ ( b + k ) Γ ( 1 - b + k - n ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition Hypergeometric0F1 b z BesselK b n 2 z 1 2 -1 n -1 2 b -1 3 2 Gamma b 1 2 1 4 4 b -1 n MeijerG 1 -1 b 2 -1 1 -1 b 2 -1 1 4 2 b 1 1 4 3 -1 2 b 1 -1 b -1 n 2 -1 b n 2 -1 1 4 2 b 1 1 4 3 -1 2 b -1 1 2 b n 1 -1 1 2 3 b n 4 z -1 b 2 -1 b -1 n 1 2 z -1 b n 2 -1 1 2 -1 k 0 n 2 -1 -1 n 1 2 -1 4 k z k Gamma k -1 n n 2 -1 1 2 Pochhammer 1 -1 k n 2 -1 n -1 n 2 -1 k Gamma b k Gamma 1 -1 b k -1 n -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Hypergeometric0F1", "[", RowBox[List["b_", ",", "z_"]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["b_", "+", "n_"]], ",", RowBox[List["2", " ", SqrtBox["z_"]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", "b"]]], " ", RowBox[List["Gamma", "[", "b", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Csc", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "+", RowBox[List["4", " ", "b"]]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "b"]], "2"], ",", RowBox[List["1", "-", FractionBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List["b", "-", "n"]], "2"]]], ",", FractionBox[RowBox[List["b", "+", "n"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["b", "+", "n"]], ")"]]]], ",", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "b"]], "+", "n"]], ")"]]]]]]]], "}"]]]], "}"]], ",", RowBox[List["4", " ", "z"]]]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Csc", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], " ", SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "b", "-", "n"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["b", "+", "n"]], ")"]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "n"]], "2"], "]"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["z", "k"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "b", "+", "k", "-", "n"]], "]"]]]]]]]]], SqrtBox["\[Pi]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02