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 BesselI

 http://functions.wolfram.com/03.02.20.0021.01

 Input Form

 D[BesselI[\[Nu], z], {z, n}] == Sum[(-1)^(m + n) Binomial[n, m] Pochhammer[-\[Nu], n - m] Sum[(((-1)^(k - 1) 2^(2 k - m) Pochhammer[-m, 2 (m - k)] Pochhammer[\[Nu], k])/(m - k)!) ((z/2) Sum[((k - j - 1)!/(j! (k - 2 j - 1)! Pochhammer[1 - k - \[Nu], j] Pochhammer[\[Nu], j + 1])) (-(z^2/4))^j BesselI[\[Nu] - 1, z], {j, 0, k - 1}] - Sum[((k - j)!/(j! (k - 2 j)! Pochhammer[1 - k - \[Nu], j] Pochhammer[\[Nu], j])) (-(z^2/4))^j BesselI[\[Nu], z], {j, 0, k}]), {k, 0, m}], {m, 0, n}]/z^n /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "n"]]], RowBox[List["Binomial", "[", RowBox[List["n", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["n", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "k"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["z", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j", "-", "1"]], ")"]], "!"]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["j", "+", "1"]]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], "j"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "z"]], "]"]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " "]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", " ", "j"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "j"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], "j"], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n I ν ( z ) z n z - n m = 0 n ( - 1 ) m + n ( n m ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["m", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - ν ) n - m TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], RowBox[List["n", "-", "m"]]], Pochhammer] k = 0 m ( - 1 ) k - 1 2 2 k - m ( - m ) 2 ( m - k ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]]]]], Pochhammer] ( ν ) k TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "k"], Pochhammer] ( m - k ) ! ( z 2 j = 0 k - 1 ( k - j - 1 ) ! j ! ( k - 2 j - 1 ) ! ( 1 - k - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "k", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν ) j + 1 TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], RowBox[List["j", "+", "1"]]], Pochhammer] ( - z 2 4 ) j I ν - 1 ( z ) - j = 0 k ( k - j ) ! j ! ( k - 2 j ) ! ( 1 - k - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "k", "-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν ) j TagBox[SubscriptBox[RowBox[List["(", "\[Nu]", ")"]], "j"], Pochhammer] ( - z 2 4 ) j I ν ( z ) ) /; n Condition z n BesselI ν z z -1 n m 0 n -1 m n Binomial n m Pochhammer -1 ν n -1 m k 0 m -1 k -1 2 2 k -1 m Pochhammer -1 m 2 m -1 k Pochhammer ν k m -1 k -1 z 2 -1 j 0 k -1 k -1 j -1 j k -1 2 j -1 Pochhammer 1 -1 k -1 ν j Pochhammer ν j 1 -1 -1 z 2 4 -1 j BesselI ν -1 z -1 j 0 k k -1 j j k -1 2 j Pochhammer 1 -1 k -1 ν j Pochhammer ν j -1 -1 z 2 4 -1 j BesselI ν z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["m", "+", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["n", "-", "m"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "m"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], "j"], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List["\[Nu]", "-", "1"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", " ", "j"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", RowBox[List["j", "+", "1"]]]], "]"]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], "j"], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", RowBox[List["2", " ", "j"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Nu]", ",", "j"]], "]"]]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "k"]], ")"]], "!"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02