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http://functions.wolfram.com/06.10.06.0009.01
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Pochhammer[a, n] \[Proportional]
Gamma[n] a + Gamma[n] (EulerGamma + PolyGamma[n]) a^2 +
Gamma[n] ((1/12) (6 EulerGamma^2 - Pi^2) + EulerGamma PolyGamma[n] +
(1/2) (PolyGamma[n]^2 + PolyGamma[1, n])) a^3 + O[a^4]
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Cell[BoxData[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "n"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "n", "]"]], " ", "a"]], "+", RowBox[List[RowBox[List["Gamma", "[", "n", "]"]], RowBox[List["(", RowBox[List["EulerGamma", " ", "+", RowBox[List["PolyGamma", "[", "n", "]"]]]], ")"]], SuperscriptBox["a", "2"]]], " ", "+", RowBox[List[RowBox[List["Gamma", "[", "n", "]"]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", SuperscriptBox["EulerGamma", "2"]]], "-", SuperscriptBox["\[Pi]", "2"]]], ")"]]]], "+", RowBox[List["EulerGamma", " ", RowBox[List["PolyGamma", "[", "n", "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["1", " "]], "2"], " ", RowBox[List["(", " ", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", "n", "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "n"]], "]"]]]], ")"]]]]]], ")"]], SuperscriptBox["a", "3"]]], " ", "+", RowBox[List["O", "[", SuperscriptBox["a", "4"], "]"]]]]]]]]
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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> <semantics> <msub> <mrow> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "n"], Pochhammer] </annotation> </semantics> <mo> ∝ </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> + </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 12 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <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> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> a </mi> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> Pochhammer </ci> <ci> a </ci> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <ci> Γ </ci> <ci> n </ci> <ci> a </ci> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <ci> n </ci> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <ci> n </ci> </apply> <eulergamma /> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 12 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <eulergamma /> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <eulergamma /> <apply> <ci> PolyGamma </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Pochhammer", "[", RowBox[List["a_", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "n", "]"]], " ", "a"]], "+", RowBox[List[RowBox[List["Gamma", "[", "n", "]"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["PolyGamma", "[", "n", "]"]]]], ")"]], " ", SuperscriptBox["a", "2"]]], "+", RowBox[List[RowBox[List["Gamma", "[", "n", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "12"], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", SuperscriptBox["EulerGamma", "2"]]], "-", SuperscriptBox["\[Pi]", "2"]]], ")"]]]], "+", RowBox[List["EulerGamma", " ", RowBox[List["PolyGamma", "[", "n", "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", "n", "]"]], "2"], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "n"]], "]"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["a", "3"]]], "+", SuperscriptBox[RowBox[List["O", "[", "a", "]"]], "4"]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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