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http://functions.wolfram.com/05.01.26.0001.01
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HermiteH[n, z] == 2^n Pi (HypergeometricPFQRegularized[{-(n/2)}, {1/2}, z^2]/
Gamma[(1 - n)/2] - (z HypergeometricPFQRegularized[{(1 - n)/2}, {3/2},
z^2])/Gamma[-(n/2)])
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Cell[BoxData[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", "n"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["n", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "n"]], "2"], "]"]]], "-", FractionBox[RowBox[List["z", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "-", "n"]], "2"], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["n", "2"]]], "]"]]]]], ")"]]]]]]]]
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<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> ⩵ </mo> <mrow> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </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["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["-", FractionBox["n", "2"]]], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[SuperscriptBox["z", "2"], Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </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["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[FractionBox[RowBox[List["1", "-", "n"]], "2"], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[SuperscriptBox["z", "2"], Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] </annotation> </semantics> </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> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <pi /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Gamma </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> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric1F1Regularized </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> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <ci> z </ci> <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> <ci> Hypergeometric1F1Regularized </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> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", "n"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["n", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "n"]], "2"], "]"]]], "-", FractionBox[RowBox[List["z", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "-", "n"]], "2"], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["n", "2"]]], "]"]]]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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