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http://functions.wolfram.com/03.21.06.0024.01
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SphericalBesselJ[\[Nu], z] == Subscript[F, Infinity][z, \[Nu]] /;
Subscript[F, n][z, \[Nu]] == (Sqrt[Pi]/2) (z/2)^\[Nu]
Sum[((-1)^k z^(2 k))/(4^k Gamma[k + \[Nu] + 3/2] k!), {k, 0, n}] ==
SphericalBesselJ[\[Nu], z] + (-1)^n 2^(-3 - 2 n - \[Nu]) Sqrt[Pi]
z^(2 + \[Nu] + 2 n) HypergeometricPFQRegularized[{1},
{2 + n, 5/2 + n + \[Nu]}, -(z^2/4)] && Element[n, Integers] && n >= 0
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[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]", RowBox[List[FractionBox["1", "2"], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[SuperscriptBox["4", "k"], " ", RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Nu]", "+", FractionBox["3", "2"]]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "n"]], "-", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["2", "+", "\[Nu]", "+", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List[FractionBox["5", "2"], "+", "n", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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