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   http://functions.wolfram.com/07.31.03.0062.01
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    HypergeometricPFQ[{-n, -(1/2) - n, Subscript[a, 3], \[Ellipsis], 
    Subscript[a, q + 1]}, {3/2, Subscript[a, 3] - 1, \[Ellipsis], 
    Subscript[a, q + 1] - 1}, -1] == 0 /; 
 Subscript[a, 3] == Subscript[a, 4] == \[Ellipsis] == Subscript[a, q + 1] == 
   1 - n/2 && Element[(q - 1)/2, Integers] && (q - 1)/2 >= 0 && 
  Element[(n - 1)/2, Integers] && (n - 1)/2 >= 0 
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   Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]], ",", SubscriptBox["a", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[SubscriptBox["a", "3"], "-", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "-", "1"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "3"], "\[Equal]", SubscriptBox["a", "4"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "\[Equal]", RowBox[List["1", "-", FractionBox["n", "2"]]]]], "\[And]", RowBox[List[FractionBox[RowBox[List["q", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[FractionBox[RowBox[List["q", "-", "1"]], "2"], "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[Element]", "Integers"]], "\[And]", RowBox[List[FractionBox[RowBox[List["n", "-", "1"]], "2"], "\[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>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <msub>  <mi> F </mi>  <mi> q </mi>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mi> n </mi>  </mrow>  <mo> , </mo>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> - </mo>  <mi> n </mi>  </mrow>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 3 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <msub>  <mi> a </mi>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  </mrow>  <mo> ; </mo>  <mrow>  <mfrac>  <mn> 3 </mn>  <mn> 2 </mn>  </mfrac>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 3 </mn>  </msub>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ; </mo>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "3"], "-", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "-", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", RowBox[List["-", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation>  </semantics>  <mo> ⩵ </mo>  <mn> 0 </mn>  </mrow>  <mo> /; </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mn> 3 </mn>  </msub>  <mo> ⩵ </mo>  <msub>  <mi> a </mi>  <mn> 4 </mn>  </msub>  <mo> ⩵ </mo>  <mo> … </mo>  <mo> ⩵ </mo>  <msub>  <mi> a </mi>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <mo> ⩵ </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mfrac>  <mi> n </mi>  <mn> 2 </mn>  </mfrac>  </mrow>  </mrow>  <mo> ∧ </mo>  <mrow>  <mfrac>  <mrow>  <mi> q </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> ∈ </mo>  <mi> ℕ </mi>  </mrow>  <mo> ∧ </mo>  <mrow>  <mfrac>  <mrow>  <mi> n </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> ∈ </mo>  <mi> ℕ </mi>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> Condition </ci>  <apply>  <eq />  <apply>  <ci> HypergeometricPFQ </ci>  <list>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> n </ci>  </apply>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 3 </cn>  </apply>  <ci> … </ci>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <apply>  <plus />  <ci> q </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  </list>  <list>  <cn type='rational'> 3 <sep /> 2 </cn>  <apply>  <plus />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 3 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  <ci> … </ci>  <apply>  <plus />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <apply>  <plus />  <ci> q </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </list>  <cn type='integer'> 1 </cn>  </apply>  <cn type='integer'> 0 </cn>  </apply>  <apply>  <and />  <apply>  <eq />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 3 </cn>  </apply>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 4 </cn>  </apply>  <ci> … </ci>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <apply>  <plus />  <ci> q </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <ci> n </ci>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <in />  <apply>  <times />  <apply>  <plus />  <ci> q </ci>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <ci> ℕ </ci>  </apply>  <apply>  <in />  <apply>  <times />  <apply>  <plus />  <ci> n </ci>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <ci> ℕ </ci>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
   
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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},{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] |  |   |  
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