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http://functions.wolfram.com/07.44.17.0018.01
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WhittakerM[\[Nu], \[Mu], z] ==
(z^(1/2 + \[Mu]) Sum[((Pochhammer[1/2 + \[Mu] - \[Nu], k] z^k)/
(k! Pochhammer[1 + 2 \[Mu], k])) HypergeometricPFQ[
{1, (k + \[Mu] - \[Nu])/n + 1/(2 n), \[Ellipsis],
(k + \[Mu] - \[Nu])/n + (n - 1)/(2 n)}, {(k + 1)/n, \[Ellipsis],
(k + n)/n, (2 \[Mu] + k + 1)/n, \[Ellipsis], (2 \[Mu] + k + n)/n},
z^n/n^n], {k, 0, n - 1}])/E^(z/2)
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Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List["WhittakerM", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], "/", "2"]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "+", "\[Mu]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Mu]", "-", "\[Nu]"]], ",", "k"]], "]"]], SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "\[Mu]"]]]], ",", "k"]], "]"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List[FractionBox[RowBox[List["k", "+", "\[Mu]", "-", "\[Nu]"]], "n"], "+", FractionBox["1", RowBox[List["2", " ", "n"]]]]], ",", "\[Ellipsis]", ",", RowBox[List[FractionBox[RowBox[List["k", "+", "\[Mu]", "-", "\[Nu]"]], "n"], "+", FractionBox[RowBox[List["n", "-", "1"]], RowBox[List["2", " ", "n"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["k", "+", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["k", "+", "n"]], "n"], ",", FractionBox[RowBox[List[RowBox[List["2", "\[Mu]"]], "+", "k", "+", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "k", "+", "n"]], "n"]]], "}"]], ",", RowBox[List[SuperscriptBox["n", RowBox[List["-", "n"]]], " ", SuperscriptBox["z", "n"]]]]], "]"]]]]]]]]]]]]]]
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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> <semantics> <mi> M </mi> <annotation encoding='Mathematica'> TagBox["M", WhittakerM] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> n </mi> <mtext> </mtext> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mi> n </mi> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mrow> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mi> n </mi> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> μ </mi> </mrow> <mo> + </mo> <mi> n </mi> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <mrow> <msup> <mi> n </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["n", " ", "+", "1"]], TraditionalForm]], SubscriptBox["F", RowBox[List["2", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox[RowBox[List["k", "+", "\[Mu]", "-", "\[Nu]"]], "n"], "+", FractionBox["1", RowBox[List["2", " ", "n"]]]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[RowBox[List[FractionBox[RowBox[List["k", "+", "\[Mu]", "-", "\[Nu]"]], "n"], "+", FractionBox[RowBox[List["n", "-", "1"]], RowBox[List["2", " ", "n"]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[FractionBox[RowBox[List["k", "+", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]]]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", RowBox[List["2", " ", "\[Mu]"]], "+", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", RowBox[List["2", " ", "\[Mu]"]], "+", "n"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ";", RowBox[List[SuperscriptBox["n", RowBox[List["-", "n"]]], " ", SuperscriptBox["z", "n"]]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> WhittakerM </ci> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> μ </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> μ </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> k </ci> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> … </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
