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http://functions.wolfram.com/07.36.03.0011.01
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AppellF1[a, Subscript[b, 1], Subscript[b, 2], a - n, Subscript[z, 1],
Subscript[z, 2]] -
Sum[((Pochhammer[-n, k + l] Pochhammer[Subscript[b, 1], k]
Pochhammer[Subscript[b, 2], l])/(Pochhammer[a - n, k + l] k! l!))
(Subscript[z, 1]/(Subscript[z, 1] - 1))^k
(Subscript[z, 2]/(Subscript[z, 2] - 1))^l, {k, 0, n}, {l, 0, n - k}]/
((1 - Subscript[z, 1])^Subscript[b, 1] (1 - Subscript[z, 2])^
Subscript[b, 2]) /; Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", RowBox[List["a", "-", "n"]], ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "1"]]], ")"]], RowBox[List["-", SubscriptBox["b", "1"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "2"]]], ")"]], RowBox[List["-", SubscriptBox["b", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "0"]], RowBox[List["n", "-", "k"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["k", "+", "l"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "1"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "2"], ",", "l"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "-", "n"]], ",", RowBox[List["k", "+", "l"]]]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["l", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["z", "1"], RowBox[List[SubscriptBox["z", "1"], "-", "1"]]], ")"]], "k"], SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["z", "2"], RowBox[List[SubscriptBox["z", "2"], "-", "1"]]], ")"]], "l"]]]]]]]]]]], "/;", 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> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ; </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> l </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> l </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], RowBox[List["k", "+", "l"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "1"], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mi> l </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "2"], ")"]], "l"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> l </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "n"]], ")"]], RowBox[List["k", "+", "l"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> l </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mi> l </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <plus /> <apply> <ci> AppellF1 </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> l </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <plus /> <ci> k </ci> <ci> l </ci> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <ci> k </ci> <ci> l </ci> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <ci> l </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> l </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["AppellF1", "[", RowBox[List["a", ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", RowBox[List["a", "-", "n"]], ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "1"]]], ")"]], RowBox[List["-", SubscriptBox["b", "1"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["z", "2"]]], ")"]], RowBox[List["-", SubscriptBox["b", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "0"]], RowBox[List["n", "-", "k"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["k", "+", "l"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "1"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "2"], ",", "l"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["z", "1"], RowBox[List[SubscriptBox["z", "1"], "-", "1"]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["z", "2"], RowBox[List[SubscriptBox["z", "2"], "-", "1"]]], ")"]], "l"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "-", "n"]], ",", RowBox[List["k", "+", "l"]]]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["l", "!"]]]]]]]]]]]]], "/;", 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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