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 BesselK

 http://functions.wolfram.com/03.04.26.0113.01

 Input Form

 BesselI[\[Nu], z] BesselK[1 + n + \[Nu], z] == (((-1)^n Pi^(3/2) Csc[(1/4) Pi ((-1)^n + 4 \[Nu])])/Sqrt[2]) MeijerG[{{0, 1/2}, {3/4 + \[Nu], 1/4}}, {{(1 + n)/2, (1 + n)/2 + \[Nu]}, {3/4 + \[Nu], 1/4, -((1 + n)/2), -((1 + n)/2) - \[Nu]}}, z, 1/2] - (-1)^n Sqrt[Pi] Csc[\[Nu] Pi] Sum[(((-1)^Floor[(1 + n)/2] z^(-1 + 2 k - n) Gamma[1/2 + k - n + Floor[n/2]])/(k! Gamma[k - n - \[Nu]] Gamma[1 + k + \[Nu]])) Pochhammer[1 - k + Floor[n/2], n - Floor[n/2]], {k, 0, Floor[n/2]}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["1", "+", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Csc", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "]"]]]], SqrtBox["2"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], ",", FractionBox["1", "4"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], ",", FractionBox["1", "4"], ",", RowBox[List["-", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "n"]], "2"], "]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "k"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "n", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "k", "+", "\[Nu]"]], "]"]]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 I ν ( z ) K n + ν + 1 ( z ) 1 2 ( ( - 1 ) n π 3 / 2 csc ( 1 4 π ( 4 ν + ( - 1 ) n ) ) ) G 4 , 6 2 , 2 ( z , 1 2 0 , 1 2 , ν + 3 4 , 1 4 n + 1 2 , n + 1 2 + ν , ν + 3 4 , 1 4 , - 1 2 ( n + 1 ) , - 1 2 ( n + 1 ) - ν ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "6"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox["z", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["3", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List["n", "+", "1"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["n", "+", "1"]], "2"], "+", "\[Nu]"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", FractionBox["3", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["1", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]]]], "-", "\[Nu]"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] - ( - 1 ) n π csc ( ν π ) k = 0 n 2 ( - 1 ) n + 1 2 z 2 k - n - 1 Γ ( k - n + n 2 + 1 2 ) ( 1 - k + n 2 ) n - n 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "k", "+", RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]]], ")"]], RowBox[List["n", "-", RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]]]], Pochhammer] k ! Γ ( k - n - ν ) Γ ( k + ν + 1 ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] Condition BesselI ν z BesselK n ν 1 z 1 2 1 2 -1 -1 n 3 2 1 4 4 ν -1 n MeijerG 0 1 2 ν 3 4 1 4 n 1 2 -1 n 1 2 -1 ν ν 3 4 1 4 -1 1 2 n 1 -1 1 2 n 1 -1 ν z 1 2 -1 -1 n 1 2 ν k 0 n 2 -1 -1 n 1 2 -1 z 2 k -1 n -1 Gamma k -1 n n 2 -1 1 2 Pochhammer 1 -1 k n 2 -1 n -1 n 2 -1 k Gamma k -1 n -1 ν Gamma k ν 1 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], " ", RowBox[List["BesselK", "[", RowBox[List[RowBox[List["1", "+", "n_", "+", "\[Nu]_"]], ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Csc", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "]"]]]], ")"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], ",", FractionBox["1", "4"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], ",", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", "\[Nu]"]], ",", FractionBox["1", "4"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], "-", "\[Nu]"]]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], SqrtBox["2"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Csc", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "n"]], "2"], "]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "k"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "n", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "k", "+", "\[Nu]"]], "]"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02