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http://functions.wolfram.com/07.20.20.0036.02
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D[(z^(2 b - 2) Hypergeometric1F1[Floor[n/2] + 1/2, b, z^2])/E^z^2, {z, n}] ==
((-1)^(n - 2 Floor[n/2]) Pochhammer[2 - 2 b, n] z^(2 b - n - 2)
Hypergeometric1F1[-n + Floor[n/2] + 1/2, b - n + Floor[n/2], z^2])/
E^z^2 /; Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["2", "b"]], "-", "2"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "+", FractionBox["1", "2"]]], ",", "b", ",", SuperscriptBox["z", "2"]]], "]"]]]], ")"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", RowBox[List["2", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", RowBox[List["2", "b"]]]], ",", "n"]], "]"]], SuperscriptBox["z", RowBox[List[RowBox[List["2", "b"]], "-", "n", "-", "2"]]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "+", FractionBox["1", "2"]]], ",", RowBox[List["b", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mi> b </mi> <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["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List[RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]], "+", FractionBox["1", "2"]]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["z", "2"], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", RowBox[List["2", " ", "b"]]]], ")"]], "n"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <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["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List[RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]], "-", "n", "+", FractionBox["1", "2"]]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["b", "-", "n", "+", RowBox[List["\[LeftFloor]", FractionBox["n", "2"], "\[RightFloor]"]]]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[SuperscriptBox["z", "2"], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric1F1 </ci> <apply> <plus /> <apply> <floor /> <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> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> <ci> n </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric1F1 </ci> <apply> <plus /> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z_", RowBox[List[RowBox[List["2", " ", "b_"]], "-", "2"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List[RowBox[List["Floor", "[", FractionBox["n_", "2"], "]"]], "+", FractionBox["1", "2"]]], ",", "b_", ",", SuperscriptBox["z_", "2"]]], "]"]]]], ")"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", RowBox[List["2", " ", "b"]]]], ",", "n"]], "]"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["2", " ", "b"]], "-", "n", "-", "2"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "+", FractionBox["1", "2"]]], ",", RowBox[List["b", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", SuperscriptBox["z", "2"]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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HypergeometricPFQ[{},{},z] | HypergeometricPFQ[{},{b},z] | HypergeometricPFQ[{a},{},z] | HypergeometricPFQ[{a1},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1},z] | HypergeometricPFQ[{a1,a2},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3},{b1,b2},z] | HypergeometricPFQ[{a1,a2,a3,a4},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5},{b1,b2,b3,b4},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5,a6},{b1,b2,b3,b4,b5},z] | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] | |
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