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 Hypergeometric0F1

 http://functions.wolfram.com/07.17.20.0017.01

 Input Form

 Derivative[1, 0][Hypergeometric0F1][1/2 - n, -z] == Gamma[1/2 - n] z^((1 + 2 n)/4) (PolyGamma[1/2 - n] - (1/2) Log[z]) BesselJ[-(1/2) - n, 2 Sqrt[z]] + (1/(2 n)!) (Sum[Binomial[n, 2 k] (2 n - 2 k)! ((PolyGamma[k + 1/2] - PolyGamma[k - n + 1/2] + CosIntegral[4 Sqrt[z]]) Cos[2 Sqrt[z]] + Sin[2 Sqrt[z]] SinIntegral[4 Sqrt[z]]) (-16 z)^k, {k, 0, Floor[n/2]}] - 4 Sqrt[z] Sum[Binomial[n, 2 k + 1] (2 n - 2 k - 1)! ((PolyGamma[k - n + 1/2] - PolyGamma[k + 3/2] - CosIntegral[4 Sqrt[z]]) Sin[2 Sqrt[z]] + Cos[2 Sqrt[z]] SinIntegral[4 Sqrt[z]]) (-16 z)^k, {k, 0, Floor[(n - 1)/2]}]) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "Hypergeometric0F1", "]"]], "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "n"]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], SuperscriptBox["z", FractionBox[RowBox[List["1", "+", RowBox[List["2", "n"]]]], "4"]], RowBox[List["(", " ", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], "-", RowBox[List[FractionBox["1", "2"], RowBox[List["Log", "[", " ", "z", "]"]]]]]], ")"]], RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["2", "n"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["2", "k"]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "-", RowBox[List["2", "k"]]]], ")"]], "!"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]], "+", RowBox[List["CosIntegral", "[", RowBox[List["4", SqrtBox["z"]]], "]"]]]], ")"]], RowBox[List["Cos", "[", RowBox[List["2", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List["2", SqrtBox["z"]]], "]"]], RowBox[List["SinIntegral", "[", RowBox[List["4", SqrtBox["z"]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "16"]], "z"]], ")"]], "k"]]]]], "-", RowBox[List["4", " ", SqrtBox["z"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], " ", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], "-", RowBox[List["CosIntegral", "[", RowBox[List["4", SqrtBox["z"]]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", SqrtBox["z"]]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["4", SqrtBox["z"]]], "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "16"]], "z"]], ")"]], "k"]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 0 F 1 ( 1 , 0 ) ( TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] ; 1 2 - n TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "n"]], Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] ; - z TagBox[RowBox[List["-", "z"]], Hypergeometric0F1, Rule[Editable, True]] ) J - n - 1 2 ( 2 z ) Γ ( 1 2 - n ) ( ψ TagBox["\[Psi]", PolyGamma] ( 1 2 - n ) - log ( z ) 2 ) z 2 n + 1 4 + 1 ( 2 n ) ! ( k = 0 n 2 ( n 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( 2 n - 2 k ) ! ( cos ( 2 z ) ( Ci ( 4 z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) + sin ( 2 z ) Si ( 4 z ) ) ( - 16 z ) k - 4 z k = 0 n - 1 2 ( n 2 k + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( 2 n - 2 k - 1 ) ! ( ( - Ci ( 4 z ) + ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) ) sin ( 2 z ) + cos ( 2 z ) Si ( 4 z ) ) ( - 16 z ) k ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] 0 F 1 ( 1 , 0 ) ( TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] ; 1 2 - n TagBox[TagBox[TagBox[RowBox[List[FractionBox["1", "2"], "-", "n"]], Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] ; - z TagBox[RowBox[List["-", "z"]], Hypergeometric0F1, Rule[Editable, True]] ) J - n - 1 2 ( 2 z ) Γ ( 1 2 - n ) ( ψ TagBox["\[Psi]", PolyGamma] ( 1 2 - n ) - log ( z ) 2 ) z 2 n + 1 4 + 1 ( 2 n ) ! ( k = 0 n 2 ( n 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( 2 n - 2 k ) ! ( cos ( 2 z ) ( Ci ( 4 z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) + sin ( 2 z ) Si ( 4 z ) ) ( - 16 z ) k - 4 z k = 0 n - 1 2 ( n 2 k + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( 2 n - 2 k - 1 ) ! ( ( - Ci ( 4 z ) + ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) ) sin ( 2 z ) + cos ( 2 z ) Si ( 4 z ) ) ( - 16 z ) k ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Hypergeometric0F1", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "n_"]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], "-", FractionBox[RowBox[List["Log", "[", "z", "]"]], "2"]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]], ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]], "+", RowBox[List["CosIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "16"]], " ", "z"]], ")"]], "k"]]]]], "-", RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], "-", RowBox[List["CosIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["4", " ", SqrtBox["z"]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "16"]], " ", "z"]], ")"]], "k"]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Contributed by

 Brychkov Yu.A. (2007)

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

 2007-05-02