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 | | http://functions.wolfram.com/07.23.03.0083.01 | 
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 | | Hypergeometric2F1[a, b, b - n, z] == (1/((1 - z)^a Pochhammer[1 - b, n])) 
   Sum[(-1)^k Pochhammer[1 - b, n - k] Pochhammer[a, k] Binomial[n, k] 
     (z/(1 - z))^k, {k, 0, n}] /; Element[n, Integers] && n >= 0 | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", "b", ",", RowBox[List["b", "-", "n"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["-", "a"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", "n"]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", RowBox[List["n", "-", "k"]]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox["z", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]]]], "/;", 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>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mn> 2 </mn>  </msub>  <msub>  <mi> F </mi>  <mn> 1 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> b </mi>  </mrow>  <mo> ; </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> n </mi>  </mrow>  <mo> ; </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["b", "-", "n"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mo> - </mo>  <mi> a </mi>  </mrow>  </msup>  <mtext>   </mtext>  </mrow>  <semantics>  <msub>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mi> n </mi>  </msub>  <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], "n"], Pochhammer] </annotation>  </semantics>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> k </mi>  <mo> = </mo>  <mn> 0 </mn>  </mrow>  <mi> n </mi>  </munderover>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mi> k </mi>  </msup>  <mo> ⁢ </mo>  <semantics>  <msub>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mi> n </mi>  <mo> - </mo>  <mi> k </mi>  </mrow>  </msub>  <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], RowBox[List["n", "-", "k"]]], Pochhammer] </annotation>  </semantics>  <mo> ⁢ </mo>  <semantics>  <msub>  <mrow>  <mo> ( </mo>  <mi> a </mi>  <mo> ) </mo>  </mrow>  <mi> k </mi>  </msub>  <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] </annotation>  </semantics>  <mo> ⁢ </mo>  <semantics>  <mrow>  <mo> ( </mo>  <mtable>  <mtr>  <mtd>  <mi> n </mi>  </mtd>  </mtr>  <mtr>  <mtd>  <mi> k </mi>  </mtd>  </mtr>  </mtable>  <mo> ) </mo>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation>  </semantics>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mfrac>  <mi> z </mi>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> z </mi>  </mrow>  </mfrac>  <mo> ) </mo>  </mrow>  <mi> k </mi>  </msup>  </mrow>  </mrow>  </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>  <ci> Hypergeometric2F1 </ci>  <ci> a </ci>  <ci> b </ci>  <apply>  <plus />  <ci> b </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> n </ci>  </apply>  </apply>  <ci> z </ci>  </apply>  <apply>  <times />  <apply>  <times />  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> z </ci>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> a </ci>  </apply>  </apply>  <apply>  <power />  <apply>  <ci> Pochhammer </ci>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  <ci> n </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <sum />  <bvar>  <ci> k </ci>  </bvar>  <lowlimit>  <cn type='integer'> 0 </cn>  </lowlimit>  <uplimit>  <ci> n </ci>  </uplimit>  <apply>  <times />  <apply>  <power />  <cn type='integer'> -1 </cn>  <ci> k </ci>  </apply>  <apply>  <ci> Pochhammer </ci>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  <apply>  <plus />  <ci> n </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> k </ci>  </apply>  </apply>  </apply>  <apply>  <ci> Pochhammer </ci>  <ci> a </ci>  <ci> k </ci>  </apply>  <apply>  <ci> Binomial </ci>  <ci> n </ci>  <ci> k </ci>  </apply>  <apply>  <power />  <apply>  <times />  <ci> z </ci>  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> z </ci>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <ci> k </ci>  </apply>  </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["Hypergeometric2F1", "[", RowBox[List["a_", ",", "b_", ",", RowBox[List["b_", "-", "n_"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", RowBox[List["n", "-", "k"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", RowBox[List["1", "-", "z"]]], ")"]], "k"]]]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", "n"]], "]"]]], "/;", 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[{a},{b},z] |  | HypergeometricPFQ[{a1},{b1,b2},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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