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 BesselJ

 http://functions.wolfram.com/03.01.06.0020.01

 Input Form

 BesselJ[\[Nu], z] \[Proportional] (-1)^n BesselJ[n, z] + ((-1)^(n - 1) Sum[((n - k - 1)!/k!) (z/2)^(2 k - n), {k, 0, n - 1}] + ((Pi (-1)^n)/2) BesselY[n, z] + ((n!/2) Sum[(1/((n - k) k!)) BesselJ[k, z] (z/2)^k, {k, 0, n - 1}])/ (-(z/2))^n + (1/n!) (-(z/2))^n Sum[(1/j) HypergeometricPFQ[{j}, {1 + j, 1 + n}, -(z^2/4)], {j, 1, n}]) (\[Nu] + n) + \[Ellipsis] /; (\[Nu] -> -n) && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["BesselJ", "[", RowBox[List["n", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], 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["\[Pi]", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], "2"], RowBox[List["BesselY", "[", RowBox[List["n", ",", "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["BesselJ", "[", RowBox[List["k", ",", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox["1", 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["\[Nu]", "+", "n"]], ")"]]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "\[Rule]", RowBox[List["-", "n"]]]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 J ν ( z ) ( - 1 ) n J n ( z ) + ( n ! 2 ( - z 2 ) - n k = 0 n - 1 1 ( n - k ) k ! J k ( z ) ( z 2 ) k + 1 n ! ( - z 2 ) 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]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["j", HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + π ( - 1 ) n 2 Y n ( z ) + ( - 1 ) n - 1 k = 0 n - 1 ( n - k - 1 ) ! k ! ( z 2 ) 2 k - n ) ( ν + n ) + /; ( ν "\[Rule]" - n ) n + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] Condition Proportional BesselJ ν z -1 n BesselJ n z n 2 -1 -1 z 2 -1 -1 n k 0 n -1 1 n -1 k k -1 BesselJ k z z 2 -1 k 1 n -1 -1 z 2 -1 n j 1 n 1 j -1 HypergeometricPFQ j j 1 n 1 -1 z 2 4 -1 -1 n 2 -1 BesselY n z -1 n -1 k 0 n -1 n -1 k -1 k -1 z 2 -1 2 k -1 n ν n Rule ν -1 n n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["BesselJ", "[", RowBox[List["n", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[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", "!"]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["BesselY", "[", RowBox[List["n", ",", "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["BesselJ", "[", RowBox[List["k", ",", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", FractionBox[RowBox[List[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["n", "!"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "n"]], ")"]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "\[Rule]", RowBox[List["-", "n"]]]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02