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 BesselI

 http://functions.wolfram.com/03.02.06.0054.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["BesselI", "[", 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[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "\[Nu]"]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]], SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", "\[Nu]"]]]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "2"], "-", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n", "+", "\[Nu]"]], "]"]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "n", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "n", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n", "+", "\[Nu]"]], ",", RowBox[List["2", "+", "n", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["BesselI", "[", 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[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], "2"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "2"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n", "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox["3", "2"], "+", "n", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n", "+", "\[Nu]"]], ",", RowBox[List["2", "+", "n", "+", RowBox[List["2", " ", "\[Nu]"]]]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "n", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "n", "+", RowBox[List["2", " ", "\[Nu]"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02