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   http://functions.wolfram.com/07.31.03.0022.01
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    HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], \[Ellipsis], 
    Subscript[a, q + 1]}, {Subscript[a, 1] - 1, Subscript[a, 2] - 1, 
    \[Ellipsis], Subscript[a, q] - 1}, z] == 
  Sum[(((k + 1) (-z)^k)/(1 - z)^(k + 2)) Sum[(-1)^j Binomial[k, j] (j + 1)^q, 
     {j, 0, k}], {k, 0, q}] /; Subscript[a, 1] == Subscript[a, 2] == 
  \[Ellipsis] == Subscript[a, q + 1] == 2 
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   Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "-", "1"]], ",", RowBox[List[SubscriptBox["a", "2"], "-", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "q"], "-", "1"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "q"], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["k", "+", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], "q"]]]]]]]]]]], "/;", RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "\[Equal]", "2"]]]]]] 
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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>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 2 </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>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mi> q </mi>  </msub>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ; </mo>  <mi> z </mi>  </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[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "-", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[TagBox[RowBox[List[SubscriptBox["a", "2"], "-", "1"]], HypergeometricPFQ, Rule[Editable, True]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "q"], "-", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> k </mi>  <mo> = </mo>  <mn> 0 </mn>  </mrow>  <mi> q </mi>  </munderover>  <mrow>  <mfrac>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> k </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mi> k </mi>  </msup>  </mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mi> k </mi>  <mo> + </mo>  <mn> 2 </mn>  </mrow>  </msup>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> j </mi>  <mo> = </mo>  <mn> 0 </mn>  </mrow>  <mi> k </mi>  </munderover>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mi> j </mi>  </msup>  <mo> ⁢ </mo>  <semantics>  <mrow>  <mo> ( </mo>  <mtable>  <mtr>  <mtd>  <mi> k </mi>  </mtd>  </mtr>  <mtr>  <mtd>  <mi> j </mi>  </mtd>  </mtr>  </mtable>  <mo> ) </mo>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation>  </semantics>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mi> j </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mi> q </mi>  </msup>  </mrow>  </mrow>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> ⩵ </mo>  <msub>  <mi> a </mi>  <mn> 2 </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>  <mn> 2 </mn>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> Condition </ci>  <apply>  <eq />  <apply>  <ci> HypergeometricPFQRegularized </ci>  <list />  <list>  <apply>  <plus />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <ci> HypergeometricPFQ </ci>  <apply>  <plus />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <ci> … </ci>  <apply>  <plus />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <ci> q </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </list>  <ci> z </ci>  </apply>  <apply>  <sum />  <bvar>  <ci> k </ci>  </bvar>  <lowlimit>  <cn type='integer'> 0 </cn>  </lowlimit>  <uplimit>  <ci> q </ci>  </uplimit>  <apply>  <times />  <apply>  <times />  <apply>  <plus />  <ci> k </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <power />  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> z </ci>  </apply>  <ci> k </ci>  </apply>  <apply>  <power />  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> z </ci>  </apply>  </apply>  <apply>  <plus />  <ci> k </ci>  <cn type='integer'> 2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <sum />  <bvar>  <ci> j </ci>  </bvar>  <lowlimit>  <cn type='integer'> 0 </cn>  </lowlimit>  <uplimit>  <ci> k </ci>  </uplimit>  <apply>  <times />  <apply>  <power />  <cn type='integer'> -1 </cn>  <ci> j </ci>  </apply>  <apply>  <ci> Binomial </ci>  <ci> k </ci>  <ci> j </ci>  </apply>  <apply>  <power />  <apply>  <plus />  <ci> j </ci>  <cn type='integer'> 1 </cn>  </apply>  <ci> q </ci>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <eq />  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <cn type='integer'> 2 </cn>  </apply>  <ci> … </ci>  <apply>  <ci> Subscript </ci>  <ci> a </ci>  <apply>  <plus />  <ci> q </ci>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <cn type='integer'> 2 </cn>  </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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