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 | | http://functions.wolfram.com/07.32.17.0004.01 | 
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 | | d HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], 
     Subscript[a, p]}, {c + 1, d, Subscript[b, 3], \[Ellipsis], 
     Subscript[b, q]}, z] - HypergeometricPFQRegularized[
   {Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, 
   {c, d + 1, Subscript[b, 3], \[Ellipsis], Subscript[b, q]}, z] + 
  (c - d) HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], 
     Subscript[a, p]}, {c + 1, d + 1, Subscript[b, 3], \[Ellipsis], 
     Subscript[b, q]}, z] == 0 | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["d", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["c", "+", "1"]], ",", "d", ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]], "-", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List["c", ",", RowBox[List["d", "+", "1"]], ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["c", "-", "d"]], ")"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["c", "+", "1"]], ",", RowBox[List["d", "+", "1"]], ",", SubscriptBox["b", "3"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]]]], "\[Equal]", "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>  <mrow>  <mi> d </mi>  <mo> ⁢ </mo>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mi> p </mi>  </msub>  <msub>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <mi> q </mi>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mi> p </mi>  </msub>  <mo> ; </mo>  <mrow>  <mrow>  <mrow>  <mi> c </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mi> d </mi>  <mo> , </mo>  <msub>  <mi> b </mi>  <mn> 3 </mn>  </msub>  </mrow>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <msub>  <mi> b </mi>  <mi> q </mi>  </msub>  </mrow>  <mo> ; </mo>  <mi> z </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["c", "+", "1"]], ",", "d", ",", SubscriptBox["b", "3"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", "z"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation>  </semantics>  </mrow>  <mo> - </mo>  <mtext>   </mtext>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mi> p </mi>  </msub>  <msub>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <mi> q </mi>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mi> p </mi>  </msub>  <mo> ; </mo>  <mi> c </mi>  </mrow>  <mo> , </mo>  <mrow>  <mi> d </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <mrow>  <msub>  <mi> b </mi>  <mn> 3 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <msub>  <mi> b </mi>  <mi> q </mi>  </msub>  </mrow>  <mo> ; </mo>  <mi> z </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]], ";", "c"]], ",", RowBox[List["d", "+", "1"]], ",", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", "z"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation>  </semantics>  <mo> + </mo>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> d </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <semantics>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mi> p </mi>  </msub>  <msub>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <mi> q </mi>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <msub>  <mi> a </mi>  <mi> p </mi>  </msub>  <mo> ; </mo>  <mrow>  <mrow>  <mrow>  <mi> c </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <mi> d </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <msub>  <mi> b </mi>  <mn> 3 </mn>  </msub>  </mrow>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <msub>  <mi> b </mi>  <mi> q </mi>  </msub>  </mrow>  <mo> ; </mo>  <mi> z </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[RowBox[List["c", "+", "1"]], ",", RowBox[List["d", "+", "1"]], ",", SubscriptBox["b", "3"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", "z"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation>  </semantics>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mn> 0 </mn>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> FormBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <ms> d </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms>  </ms>  <apply>  <ci> FormBox </ci>  <ms> p </ms>  <ci> TraditionalForm </ci>  </apply>  </apply>  <apply>  <ci> SubscriptBox </ci>  <apply>  <ci> OverscriptBox </ci>  <ms> F </ms>  <ms> ~ </ms>  </apply>  <apply>  <ci> FormBox </ci>  <ms> q </ms>  <ci> TraditionalForm </ci>  </apply>  </apply>  </list>  </apply>  <ms> ⁡ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> 1 </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> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> p </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> ; 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</ms>  <ms> z </ms>  </list>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  <apply>  <ci> Function </ci>  <ci> Null </ci>  <apply>  <ci> HoldComplete </ci>  <apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Slot </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <ci> Slot </ci>  <cn type='integer'> 2 </cn>  </apply>  <apply>  <ci> Slot </ci>  <cn type='integer'> 3 </cn>  </apply>  </apply>  </apply>  <list>  <ci> HoldAllComplete </ci>  </list>  </apply>  </apply>  </list>  </apply>  <ms> - </ms>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms>  </ms>  <apply>  <ci> FormBox </ci>  <ms> p </ms>  <ci> TraditionalForm </ci>  </apply>  </apply>  <apply>  <ci> SubscriptBox </ci>  <apply>  <ci> OverscriptBox </ci>  <ms> F </ms>  <ms> ~ </ms>  </apply>  <apply>  <ci> FormBox </ci>  <ms> q </ms>  <ci> TraditionalForm </ci>  </apply>  </apply>  </list>  </apply>  <ms> ⁡ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> 1 </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> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> p </ms>  </apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ms> ; </ms>  <ms> c </ms>  </list>  </apply>  <ms> , </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> d </ms>  <ms> + </ms>  <ms> 1 </ms>  </list>  </apply>  <ms> , </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <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>  <ms> ; </ms>  <ms> z </ms>  </list>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  <apply>  <ci> InterpretTemplate </ci>  <apply>  <ci> Function </ci>  <apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Slot </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <ci> Slot </ci>  <cn type='integer'> 2 </cn>  </apply>  <apply>  <ci> Slot </ci>  <cn type='integer'> 3 </cn>  </apply>  </apply>  </apply>  </apply>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ci> HypergeometricPFQ </ci>  </apply>  <ms> + </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> c </ms>  <ms> - </ms>  <ms> d </ms>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms>  </ms>  <apply>  <ci> FormBox </ci>  <ms> p </ms>  <ci> TraditionalForm </ci>  </apply>  </apply>  <apply>  <ci> SubscriptBox </ci>  <apply>  <ci> OverscriptBox </ci>  <ms> F </ms>  <ms> ~ </ms>  </apply>  <apply>  <ci> FormBox </ci>  <ms> q </ms>  <ci> TraditionalForm </ci>  </apply>  </apply>  </list>  </apply>  <ms> ⁡ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> 1 </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> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> p </ms>  </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> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <ms> c </ms>  <ms> + </ms>  <ms> 1 </ms>  </list>  </apply>  <ms> , </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> d </ms>  <ms> + </ms>  <ms> 1 </ms>  </list>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> 3 </ms>  </apply>  </list>  </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>  <ms> ; </ms>  <ms> z </ms>  </list>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  <apply>  <ci> InterpretTemplate </ci>  <apply>  <ci> Function </ci>  <apply>  <ci> HypergeometricPFQ </ci>  <apply>  <ci> Slot </ci>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <ci> Slot </ci>  <cn type='integer'> 2 </cn>  </apply>  <apply>  <ci> Slot </ci>  <cn type='integer'> 3 </cn>  </apply>  </apply>  </apply>  </apply>  <apply>  <ci> Rule </ci>  <ci> Editable </ci>  <true />  </apply>  </apply>  <ci> HypergeometricPFQ </ci>  </apply>  </list>  </apply>  </list>  </apply>  <ms> ⩵ </ms>  <ms> 0 </ms>  </list>  </apply>  <ci> TraditionalForm </ci>  </apply>  </annotation-xml>  </semantics>  </math> 
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