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 SphericalBesselJ

 http://functions.wolfram.com/03.21.20.0006.01

 Input Form

 Derivative[1, 0][SphericalBesselJ][-n - 1/2, z] == Sqrt[Pi/2] ((-1)^(n - 1)/Sqrt[z]) Sum[((n - k - 1)!/k!) (z/2)^(2 k - n), {k, 0, n - 1}] + (((-1)^n Pi)/2) SphericalBesselY[n - 1/2, z] + ((n!/2) Sum[(1/((n - k) k!)) SphericalBesselJ[k - 1/2, z] (z/2)^k, {k, 0, n - 1}])/(-(z/2))^n + Sqrt[Pi/2] (1/(Sqrt[z] n!)) (-(z/2))^n Sum[(1/j) HypergeometricPFQ[{j}, {1 + j, 1 + n}, -(z^2/4)], {j, 1, n}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "SphericalBesselJ", "]"]], "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], SqrtBox["z"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", "k"]], "-", "n"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "\[Pi]"]], "2"], RowBox[List["SphericalBesselY", "[", RowBox[List[RowBox[List["n", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["n", "!"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]], RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["k", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "k"]]]]]]], "+", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], FractionBox["1", RowBox[List[SqrtBox["z"], RowBox[List["n", "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[FractionBox["1", "j"], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "j", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "j"]], ",", RowBox[List["1", "+", "n"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 j TagBox["j", BesselJ] - n - 1 2 ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) n ! 2 ( - z 2 ) - n k = 0 n - 1 1 ( n - k ) k ! j k - 1 2 ( z ) ( z 2 ) k + ( - z 2 ) n π 2 1 z n ! j = 1 n 1 j 1 F 2 ( j ; j + 1 , n + 1 ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["j", 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["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] + ( - 1 ) n π 2 y n - 1 2 ( z ) + π 2 ( - 1 ) n - 1 z k = 0 n - 1 ( - k + n - 1 ) ! ( z 2 ) 2 k - n k ! /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition 1 0 Subscript BesselJ j -1 n -1 1 2 z n 2 -1 -1 z 2 -1 -1 n k 0 n -1 1 n -1 k k -1 SphericalBesselJ k -1 1 2 z z 2 -1 k -1 z 2 -1 n 2 -1 1 2 1 z 1 2 n -1 j 1 n 1 j -1 HypergeometricPFQ j j 1 n 1 -1 z 2 4 -1 -1 n 2 -1 SphericalBesselY n -1 1 2 z 2 -1 1 2 -1 n -1 z 1 2 -1 k 0 n -1 -1 k n -1 z 2 -1 2 k -1 n k -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["SphericalBesselJ", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List[RowBox[List["-", "n_"]], "-", FractionBox["1", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "-", "n"]]]]], RowBox[List["k", "!"]]]]]]], SqrtBox["z"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[Pi]"]], ")"]], " ", RowBox[List["SphericalBesselY", "[", RowBox[List[RowBox[List["n", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["k", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", FractionBox[RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], FractionBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "j", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "j"]], ",", RowBox[List["1", "+", "n"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]], "j"]]]]], RowBox[List[SqrtBox["z"], " ", RowBox[List["n", "!"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02