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   http://functions.wolfram.com/07.32.03.0052.01
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    HypergeometricPFQRegularized[{1, Subscript[a, 2], \[Ellipsis], 
    Subscript[a, q + 1]}, {Subscript[a, 2] + 2, \[Ellipsis], 
    Subscript[a, q + 1] + 2}, 1] == 
  ((-1)^q/(q - 1)!) Sum[(((2 q - 2 k - 1)! (2 Pi)^(2 k))/((q - 2 k)! (2 k)!)) 
      Abs[BernoulliB[2 k]], {k, 1, Floor[q/2]}] + 
   (-1)^(q - 1) Binomial[2 q - 1, q - 1] /; 
 Subscript[a, 2] == Subscript[a, 3] == \[Ellipsis] == Subscript[a, q + 1] == 1 
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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>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <mi> q </mi>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mn> 1 </mn>  <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>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 2 </mn>  </mrow>  </mrow>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <mo> + </mo>  <mn> 2 </mn>  </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[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[RowBox[List[TagBox[TagBox[RowBox[List["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", RowBox[List[SubscriptBox["a", "2"], "+", "2"]]]], ",", "\[Ellipsis]", ",", RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "2"]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mi> q </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mtext>    </mtext>  <semantics>  <mrow>  <mo> ( </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> q </mi>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mi> q </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mtd>  </mtr>  </mtable>  <mo> ) </mo>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "q"]], "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["q", "-", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation>  </semantics>  </mrow>  <mo> + </mo>  <mrow>  <mfrac>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mi> q </mi>  </msup>  <mtext>   </mtext>  </mrow>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> q </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ! </mo>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> k </mi>  <mo> = </mo>  <mn> 1 </mn>  </mrow>  <mrow>  <mo> ⌊ </mo>  <mfrac>  <mi> q </mi>  <mn> 2 </mn>  </mfrac>  <mo> ⌋ </mo>  </mrow>  </munderover>  <mfrac>  <mrow>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> q </mi>  </mrow>  <mo> - </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ! </mo>  </mrow>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation>  </semantics>  <msub>  <semantics>  <mi> B </mi>  <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation>  </semantics>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  </msub>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation>  </semantics>  </mrow>  </mrow>  <mrow>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> q </mi>  <mo> - </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ! </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ! </mo>  </mrow>  </mrow>  </mfrac>  </mrow>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <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>  <mo> ⩵ </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <mrow>  <mrow>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <msub>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <mi> q </mi>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mn> 1 </mn>  <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>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 2 </mn>  </mrow>  </mrow>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <mo> + </mo>  <mn> 2 </mn>  </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[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[RowBox[List[TagBox[TagBox[RowBox[List["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", RowBox[List[SubscriptBox["a", "2"], "+", "2"]]]], ",", "\[Ellipsis]", ",", RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "2"]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mi> q </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mtext>    </mtext>  <semantics>  <mrow>  <mo> ( </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> q </mi>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mi> q </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mtd>  </mtr>  </mtable>  <mo> ) </mo>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "q"]], "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["q", "-", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation>  </semantics>  </mrow>  <mo> + </mo>  <mrow>  <mfrac>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mi> q </mi>  </msup>  <mtext>   </mtext>  </mrow>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> q </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ! </mo>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> k </mi>  <mo> = </mo>  <mn> 1 </mn>  </mrow>  <mrow>  <mo> ⌊ </mo>  <mfrac>  <mi> q </mi>  <mn> 2 </mn>  </mfrac>  <mo> ⌋ </mo>  </mrow>  </munderover>  <mfrac>  <mrow>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> q </mi>  </mrow>  <mo> - </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ! </mo>  </mrow>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation>  </semantics>  <msub>  <semantics>  <mi> B </mi>  <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation>  </semantics>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  </msub>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation>  </semantics>  </mrow>  </mrow>  <mrow>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> q </mi>  <mo> - </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ! </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> k </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ! </mo>  </mrow>  </mrow>  </mfrac>  </mrow>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <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>  <mo> ⩵ </mo>  <mn> 1 </mn>  </mrow>  </mrow>  </annotation-xml>  </semantics>  </math> 
   
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