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http://functions.wolfram.com/06.01.20.0002.01
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D[n!, {n, 2}] == Gamma[n + 1] PolyGamma[n + 1]^2 +
Gamma[n + 1] PolyGamma[1, n + 1]
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Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["n", ",", "2"]], "}"]]], RowBox[List["n", "!"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["n", "+", "1"]], "]"]], "2"]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["n", "+", "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> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mn> 2 </mn> </msup> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> n </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <factorial /> <ci> n </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </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["n_", ",", "2"]], "}"]]]]], RowBox[List["n_", "!"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["n", "+", "1"]], "]"]], "2"]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["n", "+", "1"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["n", "+", "1"]]]], "]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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