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http://functions.wolfram.com/06.11.03.0014.01
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LogGamma[p/q + n] == LogGamma[p/q] - n Log[q] + Sum[Log[q k - q + p],
{k, 1, n}] /; Element[n, Integers] && n >= 0 && Element[p, Integers] &&
p > 0 && Element[q, Integers] && q > 0 && p < q
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LogGamma", "[", RowBox[List[FractionBox["p", "q"], "+", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["LogGamma", "[", FractionBox["p", "q"], "]"]], "-", RowBox[List["n", " ", RowBox[List["Log", "[", "q", "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["Log", "[", RowBox[List[RowBox[List["q", " ", "k"]], "-", "q", "+", "p"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", ">", "0"]], "\[And]", RowBox[List["q", "\[Element]", "Integers"]], "\[And]", RowBox[List["q", ">", "0"]], "\[And]", RowBox[List["p", "<", "q"]]]]]]]]
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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> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> q </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> + </mo> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> - </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> q </mi> <mo> ∈ </mo> <semantics> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> < </mo> <mi> q </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LogGamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> LogGamma </ci> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <ln /> <ci> q </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ln /> <apply> <plus /> <ci> p </ci> <apply> <times /> <ci> k </ci> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> p </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> q </ci> <integers /> </apply> <apply> <lt /> <ci> p </ci> <ci> q </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LogGamma", "[", RowBox[List[FractionBox["p_", "q_"], "+", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["LogGamma", "[", FractionBox["p", "q"], "]"]], "-", RowBox[List["n", " ", RowBox[List["Log", "[", "q", "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["Log", "[", RowBox[List[RowBox[List["q", " ", "k"]], "-", "q", "+", "p"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", ">", "0"]], "&&", RowBox[List["q", "\[Element]", "Integers"]], "&&", RowBox[List["q", ">", "0"]], "&&", RowBox[List["p", "<", "q"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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