Tricomi confluent hypergeometric function: Specific values (formula 07.33.03.0051)

HypergeometricU

$Mathematica Notation$

Hypergeometric Functions

HypergeometricU[a,b,z]

Specific values

Specialized values

For fixed z and with symbolical integers in parameters

For fixed z and a=-n, b=1/2+-m

http://functions.wolfram.com/07.33.03.0051.01

Input Form

HypergeometricU[-n, 3/2, z] == (2^(-1 - 2 n)/Sqrt[z]) HermiteH[2 n + 1, Sqrt[z]] /; Element[n, Integers] && n >= 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["-", "n"]], ",", FractionBox["3", "2"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", "n"]]]]], SqrtBox["z"]], RowBox[List["HermiteH", "[", RowBox[List[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ",", SqrtBox["z"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox["U", HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mfrac> <msup> <mn> 2 </mn> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <msqrt> <mi> z </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> H </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricU </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HermiteH </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["-", "n_"]], ",", FractionBox["3", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["HermiteH", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]], ",", SqrtBox["z"]]], "]"]]]], SqrtBox["z"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

2007-05-02