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 SphericalBesselJ

 http://functions.wolfram.com/03.21.06.0038.01

 Input Form

 SphericalBesselJ[\[Nu], z]^2 == Subscript[F, Infinity][z . \[Nu]] /; Subscript[F, n][z, \[Nu]] == ((2^(-2 - 2 \[Nu]) Pi z^(2 \[Nu]))/ Gamma[3/2 + \[Nu]]^2) Sum[((-1)^k Pochhammer[\[Nu] + 1, k] z^(2 k))/ (Pochhammer[\[Nu] + 3/2, k] Pochhammer[2 + 2 \[Nu], k] k!), {k, 0, n}] == SphericalBesselJ[\[Nu], z]^2 + (((-1)^n Sqrt[Pi] z^(2 + 2 n + 2 \[Nu]) Gamma[2 + n + \[Nu]])/2) HypergeometricPFQRegularized[{1, 2 + n + \[Nu]}, {2 + n, 5/2 + n + \[Nu], 3 + n + 2 \[Nu]}, -z^2] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "2"], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ".", "\[Nu]"]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]], "2"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]], SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", "\[Nu]"]]]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "2"], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "n", "+", "\[Nu]"]], "]"]], " "]], "2"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["2", "+", "n", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n", "+", "\[Nu]"]], ",", RowBox[List["3", "+", "n", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

 j ν ( z ) 2 F ( z . ν ) /; ( ( F n ( z , ν ) 2 - 2 ν - 2 π z 2 ν Γ ( ν + 3 2 ) 2 k = 0 n ( - 1 ) k ( ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] z 2 k ( ν + 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], ")"]], "k"], Pochhammer] ( 2 ν + 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "2"]], ")"]], "k"], Pochhammer] k ! j ν ( z ) 2 + 1 2 ( - 1 ) n π z 2 n + 2 ν + 2 Γ ( n + ν + 2 ) 2 F ~ 3 ( 1 , n + ν + 2 ; n + 2 , n + ν + 5 2 , n + 2 ν + 3 ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "\[Nu]", "+", "2"]], 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["n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "\[Nu]", "+", FractionBox["5", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", SuperscriptBox["z", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] ) Condition SphericalBesselJ ν z 2 Subscript F z ν Subscript F n z ν 2 -2 ν -2 z 2 ν Gamma ν 3 2 2 -1 k 0 n -1 k Pochhammer ν 1 k z 2 k Pochhammer ν 3 2 k Pochhammer 2 ν 2 k k -1 SphericalBesselJ ν z 2 1 2 -1 n 1 2 z 2 n 2 ν 2 Gamma n ν 2 HypergeometricPFQRegularized 1 n ν 2 n 2 n ν 5 2 n 2 ν 3 -1 z 2 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ".", "\[Nu]"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", "1"]], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], "]"]], "2"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "2"], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "n", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["2", "+", "n", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n", "+", "\[Nu]"]], ",", RowBox[List["3", "+", "n", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02