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 BesselY

 http://functions.wolfram.com/03.03.26.0168.01

 Input Form

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

 Y ν ( z ) 0 F ~ 1 ( ; - ν - n ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["0", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "n"]], Hypergeometric0F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], Hypergeometric0F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1Regularized] 2 - n - ν - 1 π ( 2 cot ( ν π ) z ν k = 0 n 2 ( - 1 ) k + n + 1 2 z 2 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 - n - ν ) Γ ( k + ν + 1 ) + ( - 1 ) n G 3 , 5 2 , 2 ( z , 1 2 1 2 ( n + ν + 1 ) , 1 2 ( n + ν + 2 ) , 1 - ν 2 n + ν 2 + 1 , - ν 2 , n + 3 ν 2 + 1 , ν 2 , 1 - ν 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["3", ",", "5"]], 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[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[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], 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["n", "+", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"], "+", "1"]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["\[Nu]", "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition BesselY ν z Hypergeometric0F1Regularized -1 ν -1 n -1 z 2 4 -1 2 -1 n -1 ν -1 1 2 -1 2 ν z ν k 0 n 2 -1 -1 k n 1 2 -1 z 2 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 -1 n -1 ν Gamma k ν 1 -1 -1 n MeijerG 1 2 n ν 1 1 2 n ν 2 1 -1 ν 2 -1 n ν 2 -1 1 -1 ν 2 -1 n 3 ν 2 -1 1 ν 2 -1 1 -1 ν 2 -1 z 1 2 n [/itex]

 Rule Form

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

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

 2007-05-02