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   http://functions.wolfram.com/07.31.03.0008.01
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    HypergeometricPFQ[{a, b, Subscript[a, 3], \[Ellipsis], Subscript[a, p]}, 
   {a + 1, b + 1, Subscript[b, 3], \[Ellipsis], Subscript[b, q]}, z] == 
  (1/(a - b)) (a HypergeometricPFQ[{b, Subscript[a, 3], \[Ellipsis], 
       Subscript[a, p]}, {b + 1, Subscript[b, 3], \[Ellipsis], 
       Subscript[b, q]}, z] - b HypergeometricPFQ[{a, Subscript[a, 3], 
       \[Ellipsis], Subscript[a, p]}, {a + 1, Subscript[b, 3], \[Ellipsis], 
       Subscript[b, q]}, z]) /; a != b 
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   Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "b", ",", SubscriptBox["a", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "+", "1"]], ",", RowBox[List["b", "+", "1"]], ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["a", "-", "b"]]], RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["b", ",", SubscriptBox["a", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["b", "+", "1"]], ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]], "-", RowBox[List["b", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", SubscriptBox["a", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "+", "1"]], ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["a", "\[NotEqual]", "b"]]]]]] 
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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>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mi> p </mi>  </msub>  <msub>  <mi> F </mi>  <mi> q </mi>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <semantics>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> b </mi>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 3 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <msub>  <mi> a </mi>  <mi> p </mi>  </msub>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[TagBox[TagBox[RowBox[List["a", ",", "b", ",", SubscriptBox["a", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] </annotation>  </semantics>  <mo> ; </mo>  <semantics>  <mrow>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <msub>  <mi> b </mi>  <mn> 3 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <msub>  <mi> b </mi>  <mi> q </mi>  </msub>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[TagBox[TagBox[RowBox[List[RowBox[List["a", "+", "1"]], ",", RowBox[List["b", "+", "1"]], ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] </annotation>  </semantics>  <mo> ; 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</ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <ms> b </ms>  <ms> + </ms>  <ms> 1 </ms>  </list>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> 3 </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <ms> … </ms>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> q </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  </list>  </apply>  <apply>  <ci> InterpretTemplate </ci>  <apply>  <ci> Function </ci>  <list>  <apply>  <ci> SlotSequence </ci>  <cn type='integer'> 1 </cn>  </apply>  </list>  </apply>  </apply>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <apply>  <ci> InterpretTemplate </ci>  <apply>  <ci> Function </ci>  <list>  <apply>  <ci> SlotSequence </ci>  <cn type='integer'> 1 </cn>  </apply>  </list>  </apply>  </apply>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> ; </ms>  <apply>  <ci> TagBox </ci>  <ms> z </ms>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  </list>  </apply>  <ms> - </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> b </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms>  </ms>  <apply>  <ci> FormBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <ms> p </ms>  <ms> - </ms>  <ms> 1 </ms>  </list>  </apply>  <ci> TraditionalForm </ci>  </apply>  </apply>  <apply>  <ci> SubscriptBox </ci>  <ms> F </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> q </ms>  <ms> - </ms>  <ms> 1 </ms>  </list>  </apply>  </apply>  </list>  </apply>  <ms> ⁡ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <ms> a </ms>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> 3 </ms>  </apply>  <ms> , </ms>  <ms> … </ms>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> p </ms>  </apply>  </list>  </apply>  <apply>  <ci> InterpretTemplate </ci>  <apply>  <ci> Function </ci>  <list>  <apply>  <ci> SlotSequence </ci>  <cn type='integer'> 1 </cn>  </apply>  </list>  </apply>  </apply>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <apply>  <ci> InterpretTemplate </ci>  <apply>  <ci> Function </ci>  <list>  <apply>  <ci> SlotSequence </ci>  <cn type='integer'> 1 </cn>  </apply>  </list>  </apply>  </apply>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> ; </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <ms> a </ms>  <ms> + </ms>  <ms> 1 </ms>  </list>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> 3 </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <ms> … </ms>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> , </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> q </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  </list>  </apply>  <apply>  <ci> InterpretTemplate </ci>  <apply>  <ci> Function </ci>  <list>  <apply>  <ci> SlotSequence </ci>  <cn type='integer'> 1 </cn>  </apply>  </list>  </apply>  </apply>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <apply>  <ci> InterpretTemplate </ci>  <apply>  <ci> Function </ci>  <list>  <apply>  <ci> SlotSequence </ci>  <cn type='integer'> 1 </cn>  </apply>  </list>  </apply>  </apply>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> ; </ms>  <apply>  <ci> TagBox </ci>  <ms> z </ms>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  </list>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  </list>  </apply>  <ms> /; </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> a </ms>  <ms> ≠ </ms>  <ms> b </ms>  </list>  </apply>  </list>  </apply>  <ci> TraditionalForm </ci>  </apply>  </annotation-xml>  </semantics>  </math> 
   
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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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