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 BesselK

 http://functions.wolfram.com/03.04.26.0100.01

 Input Form

 BesselK[\[Nu], 2 Sqrt[z]] Hypergeometric0F1Regularized[-n - \[Nu], z] == 2^(-(3/2) - n) Sqrt[Pi] z^(\[Nu]/2) ((Pi Csc[(1/4) Pi (-(-1)^n + 4 \[Nu])] MeijerG[{{(1/2) (1 + n + \[Nu]), (1/2) (2 + n + \[Nu])}, {(1/4) (1 - 2 n - 2 \[Nu]), (1/4) (3 + 2 n + 2 \[Nu])}}, {{1 + n + \[Nu]/2, -(\[Nu]/2)}, {(1/4) (1 - 2 n - 2 \[Nu]), (1/4) (3 + 2 n + 2 \[Nu]), \[Nu]/2, 1 + n + (3 \[Nu])/2}}, 4 z])/ (2^\[Nu] z^(\[Nu]/2)) - Sqrt[2] Csc[\[Nu] Pi] Sum[(((-1)^Floor[(1 + n)/2] 4^k z^k Gamma[1/2 + k - n + Floor[n/2]])/ (k! Gamma[1 + k + \[Nu]] Gamma[k - n - \[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["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]], ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", "n"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", FractionBox["\[Nu]", "2"]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "2"]]], " ", RowBox[List["Csc", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", 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[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "n", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "n", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "n", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", FractionBox["\[Nu]", "2"], ",", RowBox[List["1", "+", "n", "+", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["4", " ", "z"]]]], "]"]]]], "-", RowBox[List[SqrtBox["2"], " ", 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["4", "k"], " ", SuperscriptBox["z", "k"], " ", 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["1", "+", "k", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "n", "-", "\[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

 K ν ( 2 z ) 0 F ~ 1 ( ; - n - ν ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]], Hypergeometric0F1Regularized, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", Hypergeometric0F1Regularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric0F1Regularized] 2 - n - 3 2 π z ν / 2 ( 2 - ν π z - ν 2 csc ( 1 4 π ( 4 ν - ( - 1 ) n ) ) G 4 , 6 2 , 2 ( 4 z 1 2 ( n + ν + 1 ) , 1 2 ( n + ν + 2 ) , 1 4 ( - 2 n - 2 ν + 1 ) , 1 4 ( 2 n + 2 ν + 3 ) n + ν 2 + 1 , - ν 2 , 1 4 ( - 2 n - 2 ν + 1 ) , 1 4 ( 2 n + 2 ν + 3 ) , ν 2 , n + 3 ν 2 + 1 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "6"]], RowBox[List["2", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["4", " ", "z"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "+", "\[Nu]", "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "+", "\[Nu]", "+", "2"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[RowBox[List["n", "+", FractionBox["\[Nu]", "2"], "+", "1"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "-", RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["\[Nu]", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"], "+", "1"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] - 2 csc ( ν π ) k = 0 n 2 ( - 1 ) n + 1 2 4 k z k 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 + ν + 1 ) Γ ( k - n - ν ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition BesselK ν 2 z 1 2 Hypergeometric0F1Regularized -1 n -1 ν z 2 -1 n -1 3 2 1 2 z ν 2 -1 2 -1 ν z -1 ν 2 -1 1 4 4 ν -1 -1 n MeijerG 1 2 n ν 1 1 2 n ν 2 1 4 -2 n -1 2 ν 1 1 4 2 n 2 ν 3 n ν 2 -1 1 -1 ν 2 -1 1 4 -2 n -1 2 ν 1 1 4 2 n 2 ν 3 ν 2 -1 n 3 ν 2 -1 1 4 z -1 2 1 2 ν k 0 n 2 -1 -1 n 1 2 -1 4 k z k Gamma k -1 n n 2 -1 1 2 Pochhammer 1 -1 k n 2 -1 n -1 n 2 -1 k Gamma k ν 1 Gamma k -1 n -1 ν -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", RowBox[List["2", " ", SqrtBox["z_"]]]]], "]"]], " ", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List[RowBox[List[RowBox[List["-", "n_"]], "-", "\[Nu]_"]], ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", "n"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["\[Nu]", "/", "2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["-", "\[Nu]"]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["-", FractionBox["\[Nu]", "2"]]]], " ", RowBox[List["Csc", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", 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[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "n", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "n", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "n", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", FractionBox["\[Nu]", "2"], ",", RowBox[List["1", "+", "n", "+", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["4", " ", "z"]]]], "]"]]]], "-", RowBox[List[SqrtBox["2"], " ", 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["4", "k"], " ", SuperscriptBox["z", "k"], " ", 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["1", "+", "k", "+", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "n", "-", "\[Nu]"]], "]"]]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02