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 Hypergeometric0F1Regularized

 http://functions.wolfram.com/07.18.26.0113.01

 Input Form

 Hypergeometric0F1Regularized[b, -(z^2/4)] BesselJ[-b - n, z] == (2^(b - 1)/Sqrt[Pi]) (2 Sum[(((-1)^(k + Floor[(n + 1)/2]) z^(-b + 2 k - n) Gamma[1/2 - n + Floor[n/2] + k])/(k! Gamma[b + k] Gamma[1 - b - n + k])) Pochhammer[1 - k + Floor[n/2], n - Floor[n/2]], {k, 0, Floor[n/2]}] - (-1)^n MeijerG[{{(1 - b)/2, 1 - b/2}, {}}, {{1 + (n - b)/2}, {(b + n)/2, -((n + b)/2), 1 - (3 b + n)/2}}, z^2]) /; Element[n, Integers] && n >= 0 && Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b", ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]], RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "b"]], "-", "n"]], ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List["b", "-", "1"]]], SqrtBox["\[Pi]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", 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["n", "+", "1"]], "2"], "]"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", "k"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "+", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "+", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "b", "-", "n", "+", "k"]], "]"]]]]], " ", 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "b"]], "2"], ",", RowBox[List["1", "-", FractionBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["n", "-", "b"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["b", "+", "n"]], "2"], " ", ",", RowBox[List["-", FractionBox[RowBox[List["n", "+", "b"]], "2"]]], ",", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["3", " ", "b"]], "+", "n"]], "2"]]]]], "}"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]

 MathML Form

 0 F ~ 1 ( ; b ; - z 2 4 ) 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]], ";", TagBox[TagBox[TagBox["b", 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] J - b - n ( z ) 2 b - 1 π ( 2 k = 0 n 2 ( ( - 1 ) k + n + 1 2 z - b + 2 k - n Γ ( k - n + n 2 + 1 2 ) ) ( - k + n 2 + 1 ) n - n 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "+", RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]], "+", "1"]], ")"]], RowBox[List["n", "-", RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]]]], Pochhammer] k ! Γ ( b + k ) Γ ( - b + k - n + 1 ) - ( - 1 ) n G 2 , 4 1 , 2 ( z 2 1 - b 2 , 1 - b 2 n - b 2 + 1 , b + n 2 , - 1 2 ( b + n ) , 1 - 1 2 ( 3 b + n ) ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "4"]], RowBox[List["1", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[SuperscriptBox["z", "2"], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[FractionBox[RowBox[List["1", "-", "b"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["b", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["n", "-", "b"]], "2"], "+", "1"]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "+", "n"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["b", "+", "n"]], ")"]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "b"]], "+", "n"]], ")"]]]]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] - π 2 < arg ( z ) π 2 Condition Hypergeometric0F1Regularized b -1 z 2 4 -1 BesselJ -1 b -1 n z 2 b -1 1 2 -1 2 k 0 n 2 -1 -1 k n 1 2 -1 z -1 b 2 k -1 n Gamma k -1 n n 2 -1 1 2 Pochhammer -1 k n 2 -1 1 n -1 n 2 -1 k Gamma b k Gamma -1 b k -1 n 1 -1 -1 -1 n MeijerG 1 -1 b 2 -1 1 -1 b 2 -1 n -1 b 2 -1 1 b n 2 -1 -1 1 2 b n 1 -1 1 2 3 b n z 2 n Inequality -1 2 -1 z 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b_", ",", RowBox[List["-", FractionBox[SuperscriptBox["z_", "2"], "4"]]]]], "]"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "b_"]], "-", "n_"]], ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["b", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", 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["n", "+", "1"]], "2"], "]"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "k"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "+", "k"]], "]"]]]], ")"]], " ", 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["b", "+", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "b", "-", "n", "+", "k"]], "]"]]]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "b"]], "2"], ",", RowBox[List["1", "-", FractionBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "+", FractionBox[RowBox[List["n", "-", "b"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["b", "+", "n"]], "2"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["n", "+", "b"]], ")"]]]], ",", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "b"]], "+", "n"]], ")"]]]]]]]], "}"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]]]

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

 2007-05-02