  
| 
 | 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   http://functions.wolfram.com/07.25.20.0011.01
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
 | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
    Derivative[{0, 0}, {0, 1}, 0][HypergeometricPFQ][
  {Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, 
  z] == (-((z Subscript[a, 1] Subscript[a, 2])/(Subscript[b, 2]^2 
     Subscript[b, 1]))) HypergeometricPFQ[
   {{1 + Subscript[a, 1], 1 + Subscript[a, 2]}, {1}, {1, Subscript[b, 2]}}, 
   {{2, 1 + Subscript[b, 1], 1 + Subscript[b, 2]}, {}, 
    {1 + Subscript[b, 2]}}, z, z] 
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], ",", "0"]], "]"]], "[", "HypergeometricPFQ", "]"]], "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["z", " ", SubscriptBox["a", "1"], SubscriptBox["a", "2"]]], RowBox[List[SubsuperscriptBox["b", "2", "2"], " ", SubscriptBox["b", "1"], " "]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", SubscriptBox["b", "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["1", "+", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "+", SubscriptBox["b", "2"]]], "}"]]]], "}"]], ",", "z", ",", "z"]], "]"]]]]]]]] 
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
  
  | 
 
  | 
   
   <math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'>  <semantics>  <mrow>  <mrow>  <msub>  <mn> 2 </mn>  </msub>  <msubsup>  <mi> F </mi>  <mn> 2 </mn>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> { </mo>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mo> } </mo>  </mrow>  <mo> , </mo>  <mrow>  <mo> { </mo>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mn> 1 </mn>  </mrow>  <mo> } </mo>  </mrow>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <semantics>  <mo>   </mo>  <annotation encoding='Mathematica'> TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⁢ </mo>  <semantics>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] </annotation>  </semantics>  </mrow>  <mo> ; </mo>  <semantics>  <mrow>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] </annotation>  </semantics>  <mo> ; </mo>  <semantics>  <mi> z </mi>  <annotation encoding='Mathematica'> TagBox["z", HypergeometricPFQ, Rule[Editable, True]] </annotation>  </semantics>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mrow>  <mi> z </mi>  <mo> ⁢ </mo>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> ⁢ </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mrow>  <msubsup>  <mi> b </mi>  <mn> 2 </mn>  <mn> 2 </mn>  </msubsup>  <mo> ⁢ </mo>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mtext>   </mtext>  </mrow>  </mfrac>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msubsup>  <mi> F </mi>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mn> 0 </mn>  <mo> ⁢ </mo>  <mn> 1 </mn>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mn> 1 </mn>  <mo> ⁢ </mo>  <mn> 2 </mn>  </mrow>  </msubsup>  <mo> ( </mo>  <mrow>  <mrow>  <mtable>  <mtr>  <mtd>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ; </mo>  <mn> 1 </mn>  <mo> ; </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  <mo> ; </mo>  </mrow>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mrow>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <mrow>  <mrow>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ; </mo>  </mrow>  <mo> ; </mo>  <mrow>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ; </mo>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> , </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <mrow>  <mrow>  <msub>  <mn> 2 </mn>  </msub>  <msubsup>  <mi> F </mi>  <mn> 2 </mn>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> { </mo>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mo> } </mo>  </mrow>  <mo> , </mo>  <mrow>  <mo> { </mo>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mn> 1 </mn>  </mrow>  <mo> } </mo>  </mrow>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <semantics>  <mo>   </mo>  <annotation encoding='Mathematica'> TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⁢ </mo>  <semantics>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] </annotation>  </semantics>  </mrow>  <mo> ; </mo>  <semantics>  <mrow>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] </annotation>  </semantics>  <mo> ; </mo>  <semantics>  <mi> z </mi>  <annotation encoding='Mathematica'> TagBox["z", HypergeometricPFQ, Rule[Editable, True]] </annotation>  </semantics>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mrow>  <mi> z </mi>  <mo> ⁢ </mo>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> ⁢ </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  </mrow>  <mrow>  <msubsup>  <mi> b </mi>  <mn> 2 </mn>  <mn> 2 </mn>  </msubsup>  <mo> ⁢ </mo>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mtext>   </mtext>  </mrow>  </mfrac>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msubsup>  <mi> F </mi>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mn> 0 </mn>  <mo> ⁢ </mo>  <mn> 1 </mn>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mn> 1 </mn>  <mo> ⁢ </mo>  <mn> 2 </mn>  </mrow>  </msubsup>  <mo> ( </mo>  <mrow>  <mrow>  <mtable>  <mtr>  <mtd>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <mrow>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ; </mo>  <mn> 1 </mn>  <mo> ; </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  <mo> ; </mo>  </mrow>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mrow>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <mrow>  <mrow>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ; </mo>  </mrow>  <mo> ; </mo>  <mrow>  <msub>  <mi> b </mi>  <mn> 2 </mn>  </msub>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ; </mo>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> , </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </annotation-xml>  </semantics>  </math> 
   
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
 
  |   
  |  
  |  
  |   
  |  |  
  |   
  |  
  |  
  |   
  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["HypergeometricPFQ", TagBox[RowBox[List["(", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", SubscriptBox["a_", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", " ", SubscriptBox["aa", "1"], " ", SubscriptBox["aa", "2"]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "2"]]]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", SubscriptBox["bb", "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["1", "+", SubscriptBox["bb", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["bb", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "+", SubscriptBox["bb", "2"]]], "}"]]]], "}"]], ",", "z", ",", "z"]], "]"]]]], RowBox[List[SubsuperscriptBox["bb", "2", "2"], " ", SubscriptBox["bb", "1"]]]]]]]]]]  |  
  |  
  |   
  |  
  |  
  | 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   Date Added to functions.wolfram.com (modification date)
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
 | 
  | 
  
| 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,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] |  | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] |  |   |  
  |  
  
  
  
 |  
 
 |