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   http://functions.wolfram.com/07.32.03.0026.01
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    HypergeometricPFQRegularized[{1, Subscript[a, 2], \[Ellipsis], 
    Subscript[a, q + 1]}, {Subscript[a, 2] + 1, \[Ellipsis], 
    Subscript[a, q + 1] + 1}, 1] == 
  (-(1/Product[Gamma[Subscript[a, j]], {j, 2, q + 1}])) 
   Sum[Product[If[l == k, 1, PolyGamma[Subscript[a, k]]/
       (Subscript[a, l] - Subscript[a, k])], {l, 2, q + 1}], 
    {k, 2, q + 1}] /; Subscript[a, l] != Subscript[a, k] && 
  2 <= l <= q + 1 && 2 <= k <= q + 1 && l != k 
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   Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["l", "\[Equal]", "k"]], ",", "1", ",", FractionBox[RowBox[List["PolyGamma", "[", SubscriptBox["a", "k"], "]"]], RowBox[List[SubscriptBox["a", "l"], "-", SubscriptBox["a", "k"]]]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "l"], "\[NotEqual]", SubscriptBox["a", "k"]]], "\[And]", RowBox[List["2", "\[LessEqual]", "l", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["2", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["l", "\[NotEqual]", "k"]]]]]]]] 
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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>  <mn> 1 </mn>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <mo> ; </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 1 </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> 1 </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["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], ";", RowBox[List[SubscriptBox["a", "2"], "+", "1"]]]], ",", "\[Ellipsis]", ",", RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]], ";", 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>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mrow>  <munderover>  <mo> ∏ </mo>  <mrow>  <mi> j </mi>  <mo> = </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </munderover>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <msub>  <mi> a </mi>  <mi> j </mi>  </msub>  <mo> ) </mo>  </mrow>  </mrow>  </mfrac>  </mrow>  <mo> ⁢ </mo>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> k </mi>  <mo> = </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </munderover>  <mrow>  <munderover>  <mo> ∏ </mo>  <munder>  <mrow>  <mi> l </mi>  <mo> = </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mi> l </mi>  <mo> ≠ </mo>  <mi> k </mi>  </mrow>  </munder>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </munderover>  <mfrac>  <mrow>  <semantics>  <mi> ψ </mi>  <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation>  </semantics>  <mo> ( </mo>  <msub>  <mi> a </mi>  <mi> k </mi>  </msub>  <mo> ) </mo>  </mrow>  <mrow>  <msub>  <mi> a </mi>  <mi> l </mi>  </msub>  <mo> - </mo>  <msub>  <mi> a </mi>  <mi> k </mi>  </msub>  </mrow>  </mfrac>  </mrow>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mi> l </mi>  </msub>  <mo> ≠ </mo>  <msub>  <mi> a </mi>  <mi> k </mi>  </msub>  </mrow>  <mo> ∧ </mo>  <mrow>  <mn> 2 </mn>  <mo> ≤ </mo>  <mi> l </mi>  <mo> ≤ </mo>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ∧ </mo>  <mrow>  <mn> 2 </mn>  <mo> ≤ </mo>  <mi> k </mi>  <mo> ≤ </mo>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ∧ </mo>  <mrow>  <mi> l </mi>  <mo> ≠ </mo>  <mi> k </mi>  </mrow>  </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>  <mn> 1 </mn>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <mo> ; </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 1 </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> 1 </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["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], ";", RowBox[List[SubscriptBox["a", "2"], "+", "1"]]]], ",", "\[Ellipsis]", ",", RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]], ";", 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>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mrow>  <munderover>  <mo> ∏ </mo>  <mrow>  <mi> j </mi>  <mo> = </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </munderover>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <msub>  <mi> a </mi>  <mi> j </mi>  </msub>  <mo> ) </mo>  </mrow>  </mrow>  </mfrac>  </mrow>  <mo> ⁢ </mo>  <mrow>  <munderover>  <mo> ∑ </mo>  <mrow>  <mi> k </mi>  <mo> = </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </munderover>  <mrow>  <munderover>  <mo> ∏ </mo>  <munder>  <mrow>  <mi> l </mi>  <mo> = </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mi> l </mi>  <mo> ≠ </mo>  <mi> k </mi>  </mrow>  </munder>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </munderover>  <mfrac>  <mrow>  <semantics>  <mi> ψ </mi>  <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation>  </semantics>  <mo> ( </mo>  <msub>  <mi> a </mi>  <mi> k </mi>  </msub>  <mo> ) </mo>  </mrow>  <mrow>  <msub>  <mi> a </mi>  <mi> l </mi>  </msub>  <mo> - </mo>  <msub>  <mi> a </mi>  <mi> k </mi>  </msub>  </mrow>  </mfrac>  </mrow>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mi> l </mi>  </msub>  <mo> ≠ </mo>  <msub>  <mi> a </mi>  <mi> k </mi>  </msub>  </mrow>  <mo> ∧ </mo>  <mrow>  <mn> 2 </mn>  <mo> ≤ </mo>  <mi> l </mi>  <mo> ≤ </mo>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ∧ </mo>  <mrow>  <mn> 2 </mn>  <mo> ≤ </mo>  <mi> k </mi>  <mo> ≤ </mo>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ∧ </mo>  <mrow>  <mi> l </mi>  <mo> ≠ </mo>  <mi> k </mi>  </mrow>  </mrow>  </mrow>  </annotation-xml>  </semantics>  </math> 
   
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  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["a_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", RowBox[List["q_", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a_", "2"], "+", "1"]], ",", "\[Ellipsis]_", ",", RowBox[List[SubscriptBox["a_", RowBox[List["q_", "+", "1"]]], "+", "1"]]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["l", "\[Equal]", "k"]], ",", "1", ",", FractionBox[RowBox[List["PolyGamma", "[", SubscriptBox["a", "k"], "]"]], RowBox[List[SubscriptBox["aa", "l"], "-", SubscriptBox["a", "k"]]]]]], "]"]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["aa", "l"], "\[NotEqual]", SubscriptBox["a", "k"]]], "&&", RowBox[List["2", "\[LessEqual]", "l", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["2", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["l", "\[NotEqual]", "k"]]]]]]]]]]  |  
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   Date Added to functions.wolfram.com (modification date)
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