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 | | http://functions.wolfram.com/07.27.03.0032.01 | 
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 | | HypergeometricPFQ[{a, b, a + 1/2}, {a + 3/2, a + 2}, 1] == 
  (2 a + 1) (a + 1) Gamma[1 - b] (Gamma[a + 1]/Gamma[a - b + 1] - 
    2 a (Gamma[a + 1/2]/Gamma[a - b + 3/2]) + 
    a (Gamma[a + 1]/Gamma[a - b + 2])) /; b != 1 && b != 2 && Re[b] < 3 | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "b", ",", RowBox[List["a", "+", RowBox[List["1", "/", "2"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "+", RowBox[List["3", "/", "2"]]]], ",", RowBox[List["a", "+", "2"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "a"]], "+", "1"]], ")"]], RowBox[List["(", RowBox[List["a", "+", "1"]], ")"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "-", "b", "+", "1"]], "]"]]], "-", RowBox[List["2", "a", FractionBox[RowBox[List["Gamma", "[", RowBox[List["a", "+", RowBox[List["1", "/", "2"]]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "-", "b", "+", RowBox[List["3", "/", "2"]]]], "]"]]]]], "+", RowBox[List["a", FractionBox[RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "-", "b", "+", "2"]], "]"]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["b", "\[NotEqual]", "1"]], "\[And]", RowBox[List["b", "\[NotEqual]", "2"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "b", "]"]], "<", "3"]]]]]]]] | 
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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> 3 </mn>  </msub>  <msub>  <mi> F </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> b </mi>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </mrow>  <mo> ; </mo>  <mrow>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mfrac>  <mn> 3 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mn> 2 </mn>  </mrow>  </mrow>  <mo> ; </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["b", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", FractionBox["1", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", FractionBox["3", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mfrac>  <mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> + </mo>  <mn> 2 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mfrac>  <mo> - </mo>  <mfrac>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> + </mo>  <mfrac>  <mn> 3 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  </mfrac>  <mo> + </mo>  <mfrac>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mrow>  <mi> b </mi>  <mo> ≠ </mo>  <mn> 1 </mn>  </mrow>  <mo> ∧ </mo>  <mrow>  <mi> b </mi>  <mo> ≠ </mo>  <mn> 2 </mn>  </mrow>  <mo> ∧ </mo>  <mrow>  <mrow>  <mi> Re </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> b </mi>  <mo> ) </mo>  </mrow>  <mo> < </mo>  <mn> 3 </mn>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> Condition </ci>  <apply>  <eq />  <apply>  <ci> HypergeometricPFQ </ci>  <list>  <ci> a </ci>  <ci> b </ci>  <apply>  <plus />  <ci> a </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </list>  <list>  <apply>  <plus />  <ci> a </ci>  <cn type='rational'> 3 <sep /> 2 </cn>  </apply>  <apply>  <plus />  <ci> a </ci>  <cn type='integer'> 2 </cn>  </apply>  </list>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <times />  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> a </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <plus />  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <ci> a </ci>  <apply>  <power />  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  <cn type='integer'> 2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> a </ci>  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <ci> a </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  <cn type='rational'> 3 <sep /> 2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <and />  <apply>  <neq />  <ci> b </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <neq />  <ci> b </ci>  <cn type='integer'> 2 </cn>  </apply>  <apply>  <lt />  <apply>  <real />  <ci> b </ci>  </apply>  <cn type='integer'> 3 </cn>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "b_", ",", RowBox[List["a_", "+", FractionBox["1", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a_", "+", FractionBox["3", "2"]]], ",", RowBox[List["a_", "+", "2"]]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "-", "b", "+", "1"]], "]"]]], "-", FractionBox[RowBox[List["2", " ", "a", " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", FractionBox["1", "2"]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["a", "-", "b", "+", FractionBox["3", "2"]]], "]"]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["a", "-", "b", "+", "2"]], "]"]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["b", "\[NotEqual]", "1"]], "&&", RowBox[List["b", "\[NotEqual]", "2"]], "&&", RowBox[List[RowBox[List["Re", "[", "b", "]"]], "<", "3"]]]]]]]]]] | 
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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},z] |  | HypergeometricPFQ[{a1,a2},{b1,b2},z] |  | HypergeometricPFQ[{a1,a2},{b1,b2,b3},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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