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variants of this functions
HermiteH






Mathematica Notation

Traditional Notation









Polynomials > HermiteH[n,z] > Representations through more general functions > Through hypergeometric functions > Involving hypergeometric U





http://functions.wolfram.com/05.01.26.0009.01









  


  










Input Form





HermiteH[n, z] == (2^n/Gamma[-(n/2)]) (2 Sqrt[Pi] (Sqrt[z^2] - z) HypergeometricPFQ[{(1 - n)/2}, {3/2}, z^2] + Sqrt[z^2] Gamma[-(n/2)] HypergeometricU[(1 - n)/2, 3/2, z^2])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "n"], " "]], RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["n", "2"]]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], "-", "z"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "-", "n"]], "2"], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["n", "2"]]], "]"]], " ", RowBox[List["HypergeometricU", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", FractionBox["3", "2"], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;1&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;n&quot;]], &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[FractionBox[&quot;3&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[SuperscriptBox[&quot;z&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox[&quot;U&quot;, HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HermiteH </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <cn type='rational'> 3 <sep /> 2 </cn> </list> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricU </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], "-", "z"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "-", "n"]], "2"], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["n", "2"]]], "]"]], " ", RowBox[List["HypergeometricU", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", FractionBox["3", "2"], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["n", "2"]]], "]"]]]]]]]










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





2001-10-29