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http://functions.wolfram.com/07.33.03.0019.01
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HypergeometricU[n, b, z] == (1/((n - 1)! Pochhammer[2 - b, n - 1]))
D[z^(n - b) E^z Gamma[b - 1, z], {z, n - 1}] /;
Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricU", "[", RowBox[List["n", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", RowBox[List["n", "-", "1"]]]], "]"]]]]], RowBox[List["D", "[", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["n", "-", "b"]]], SuperscriptBox["\[ExponentialE]", "z"], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["b", "-", "1"]], ",", "z"]], "]"]]]], ",", RowBox[List["{", RowBox[List["z", ",", RowBox[List["n", "-", "1"]]]], "}"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]
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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> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox["U", HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]], RowBox[List["n", "-", "1"]]], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mi> z </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricU </ci> <ci> n </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </degree> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <ci> z </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricU", "[", RowBox[List["n_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z", ",", RowBox[List["n", "-", "1"]]]], "}"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["n", "-", "b"]]], " ", SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["b", "-", "1"]], ",", "z"]], "]"]]]], ")"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", RowBox[List["n", "-", "1"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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