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http://functions.wolfram.com/05.01.23.0011.01
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Sum[(HermiteH[n, x + n y] t^n)/n!, {n, 0, Infinity}] ==
E^(2 x \[Alpha] - \[Alpha]^2)/(1 - 2 y \[Alpha]) /;
\[Alpha] == -(ProductLog[-2 t y]/(2 y)) && -1 < t < 1 && -1 < y < 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n", ",", RowBox[List["x", "+", RowBox[List["n", " ", "y"]]]]]], "]"]], " ", SuperscriptBox["t", "n"]]], RowBox[List["n", "!"]]]]], "\[Equal]", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "x", " ", "\[Alpha]"]], "-", SuperscriptBox["\[Alpha]", "2"]]]], RowBox[List["1", "-", RowBox[List["2", " ", "y", " ", "\[Alpha]"]]]]]]], "/;", RowBox[List[RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["-", FractionBox[RowBox[List["ProductLog", "[", RowBox[List[RowBox[List["-", "2"]], " ", "t", " ", "y"]], "]"]], RowBox[List["2", " ", "y"]]]]]]], "\[And]", RowBox[List[RowBox[List["-", "1"]], "<", "t", "<", "1"]], "\[And]", RowBox[List[RowBox[List["-", "1"]], "<", "y", "<", "1"]]]]]]]]
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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> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> t </mi> <mi> n </mi> </msup> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> ⩵ </mo> <mfrac> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mo> - </mo> <msup> <mi> α </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> y </mi> <mo> ⁢ </mo> <mi> α </mi> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> α </mi> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> W </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> t </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> < </mo> <mi> t </mi> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> < </mo> <mi> y </mi> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> HermiteH </ci> <ci> n </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <ci> n </ci> <ci> y </ci> </apply> </apply> </apply> <apply> <power /> <ci> t </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> <ci> α </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> α </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> <ci> α </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> W </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> t </ci> <ci> y </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> t </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> y </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n_", ",", RowBox[List["x_", "+", RowBox[List["n_", " ", "y_"]]]]]], "]"]], " ", SuperscriptBox["t_", "n_"]]], RowBox[List["n_", "!"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "x", " ", "\[Alpha]"]], "-", SuperscriptBox["\[Alpha]", "2"]]]], RowBox[List["1", "-", RowBox[List["2", " ", "y", " ", "\[Alpha]"]]]]], "/;", RowBox[List[RowBox[List["\[Alpha]", "\[Equal]", RowBox[List["-", FractionBox[RowBox[List["ProductLog", "[", RowBox[List[RowBox[List["-", "2"]], " ", "t", " ", "y"]], "]"]], RowBox[List["2", " ", "y"]]]]]]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", "t", "<", "1"]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", "y", "<", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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