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 SphericalBesselJ

 http://functions.wolfram.com/03.21.17.0003.01

 Input Form

 SphericalBesselJ[\[Nu], z] == (2^(-1 + n) Pochhammer[3/2 + \[Nu], -1 + n] ((1 + 2 n + 2 \[Nu]) Sum[((n - k)! SphericalBesselJ[n + \[Nu], z] z^(2 k))/ (2^(2 k) (k! (n - 2 k)! Pochhammer[-(1/2) - n - \[Nu], k] Pochhammer[3/2 + \[Nu], k])), {k, 0, Floor[n/2]}] - z Sum[((n - k - 1)! SphericalBesselJ[1 + n + \[Nu], z] z^(2 k))/ (2^(2 k) (k! (n - 2 k - 1)! Pochhammer[1/2 - n - \[Nu], k] Pochhammer[3/2 + \[Nu], k])), {k, 0, Floor[(n - 1)/2]}]))/z^n

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["n", "+", "\[Nu]"]], ",", "z"]], "]"]], SuperscriptBox["z", RowBox[List["2", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", "k"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]]]]]]], "-", RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["1", "+", "n", "+", "\[Nu]"]], ",", "z"]], "]"]], SuperscriptBox["z", RowBox[List["2", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "n", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]]]]]]]]], ")"]]]]]]]]

 MathML Form

 j ν ( z ) 2 n - 1 z - n ( ν + 3 2 ) n - 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["n", "-", "1"]]], Pochhammer] ( ( 2 n + 2 ν + 1 ) k = 0 n 2 2 - 2 k ( n - k ) ! j n + ν ( z ) z 2 k k ! ( n - 2 k ) ! ( - n - ν - 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]", "-", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( ν + 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], ")"]], "k"], Pochhammer] - z k = 0 n - 1 2 2 - 2 k ( n - k - 1 ) ! j n + ν + 1 ( z ) z 2 k k ! ( n - 2 k - 1 ) ! ( 1 2 - n - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "n", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( ν + 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["3", "2"]]], ")"]], "k"], Pochhammer] ) SphericalBesselJ ν z 2 n -1 z -1 n Pochhammer ν 3 2 n -1 2 n 2 ν 1 k 0 n 2 -1 2 -2 k n -1 k SphericalBesselJ n ν z z 2 k k n -1 2 k Pochhammer -1 n -1 ν -1 1 2 k Pochhammer ν 3 2 k -1 -1 z k 0 n -1 2 -1 2 -2 k n -1 k -1 SphericalBesselJ n ν 1 z z 2 k k n -1 2 k -1 Pochhammer 1 2 -1 n -1 ν k Pochhammer ν 3 2 k -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["n", "+", "\[Nu]"]], ",", "z"]], "]"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]]]]]]], "-", RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["1", "+", "n", "+", "\[Nu]"]], ",", "z"]], "]"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "n", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]]]]]]]]]]], ")"]]]]]]]]

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

 2007-05-02