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   http://functions.wolfram.com/07.32.20.0015.01
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    D[z^\[Alpha] HypergeometricPFQRegularized[{-n, Subscript[a, 2], \[Ellipsis], 
      Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z], 
   {z, n}] == Gamma[\[Alpha] + 1] z^(\[Alpha] - n) 
   HypergeometricPFQRegularized[{-n, \[Alpha] + 1, Subscript[a, 2], 
     \[Ellipsis], Subscript[a, p]}, {\[Alpha] - n + 1, Subscript[b, 1], 
     \[Ellipsis], Subscript[b, q]}, z] /; Element[n, Integers] && n > 0 
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   Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]], ")"]]]], "\[Equal]", " ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "+", "1"]], "]"]], SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "n"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["\[Alpha]", "+", "1"]], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["\[Alpha]", "-", "n", "+", "1"]], ",", SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "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>  <mfrac>  <mrow>  <msup>  <mo> ∂ </mo>  <mi> n </mi>  </msup>  <mrow>  <mo> ( </mo>  <mrow>  <msup>  <mi> z </mi>  <mi> α </mi>  </msup>  <mo> ⁢ </mo>  <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>  <mrow>  <semantics>  <mrow>  <mrow>  <mo> - </mo>  <mi> n </mi>  </mrow>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <msub>  <mi> a </mi>  <mi> p </mi>  </msub>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List["-", "n"]], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] </annotation>  </semantics>  <mo> ; </mo>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  </mrow>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <msub>  <mi> b </mi>  <mi> q </mi>  </msub>  <mo> ; </mo>  <semantics>  <mi> z </mi>  <annotation encoding='Mathematica'> TagBox["z", HypergeometricPFQ, Rule[Editable, True]] </annotation>  </semantics>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mrow>  <mo> ∂ </mo>  <msup>  <mi> z </mi>  <mi> n </mi>  </msup>  </mrow>  </mfrac>  <mo> ⩵ </mo>  <mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> α </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mi> α </mi>  <mo> - </mo>  <mi> n </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mrow>  <msub>  <mo>   </mo>  <mrow>  <mi> p </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  <msub>  <mover>  <mi> F </mi>  <mo> ~ </mo>  </mover>  <mrow>  <mi> q </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msub>  </mrow>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <semantics>  <mrow>  <mrow>  <mo> - </mo>  <mi> n </mi>  </mrow>  <mo> , </mo>  <mrow>  <mi> α </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <msub>  <mi> a </mi>  <mn> 2 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <msub>  <mi> a </mi>  <mi> p </mi>  </msub>  </mrow>  <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["\[Alpha]", "+", "1"]], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] </annotation>  </semantics>  <mo> ; </mo>  <mrow>  <mi> α </mi>  <mo> - </mo>  <mi> n </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> , </mo>  <msub>  <mi> b </mi>  <mn> 1 </mn>  </msub>  <mo> , </mo>  <mo> … </mo>  <mo> , </mo>  <mrow>  <msub>  <mi> b </mi>  <mi> q </mi>  </msub>  <mo> ; </mo>  <semantics>  <mi> z </mi>  <annotation encoding='Mathematica'> TagBox["z", HypergeometricPFQ, Rule[Editable, True]] </annotation>  </semantics>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> n </mi>  <mo> ∈ </mo>  <msup>  <mi> ℕ </mi>  <mo> + </mo>  </msup>  </mrow>  </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>  <ms> [ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SuperscriptBox </ci>  <ms> z </ms>  <ms> α </ms>  </apply>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <apply>  <ci> ErrorBox </ci>  <ms>  </ms>  </apply>  <ms> p </ms>  </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> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <ms> - </ms>  <ms> n </ms>  </list>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> 2 </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>  <ms> ; </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> 1 </ms>  </apply>  </list>  </apply>  <ms> , </ms>  <ms> … </ms>  <ms> , </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> q </ms>  </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>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  <ms> , </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> { </ms>  <ms> z </ms>  <ms> , </ms>  <ms> n </ms>  <ms> } </ms>  </list>  </apply>  <ms> ] </ms>  </list>  </apply>  <ms> ⩵ </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <ms> Γ </ms>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> α </ms>  <ms> + </ms>  <ms> 1 </ms>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  <apply>  <ci> SuperscriptBox </ci>  <ms> z </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> α </ms>  <ms> - </ms>  <ms> n </ms>  </list>  </apply>  </apply>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms>  </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> p </ms>  <ms> + </ms>  <ms> 1 </ms>  </list>  </apply>  </apply>  <apply>  <ci> SubscriptBox </ci>  <apply>  <ci> OverscriptBox </ci>  <ms> F </ms>  <ms> ~ </ms>  </apply>  <apply>  <ci> FormBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <ms> q </ms>  <ms> + </ms>  <ms> 1 </ms>  </list>  </apply>  <ci> TraditionalForm </ci>  </apply>  </apply>  </list>  </apply>  <ms> ⁡ </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> ( </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> TagBox </ci>  <apply>  <ci> TagBox </ci>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> RowBox </ci>  <list>  <ms> - </ms>  <ms> n </ms>  </list>  </apply>  <ms> , </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> α </ms>  <ms> + </ms>  <ms> 1 </ms>  </list>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> a </ms>  <ms> 2 </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>  <ms> ; </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> α </ms>  <ms> - </ms>  <ms> n </ms>  <ms> + </ms>  <ms> 1 </ms>  </list>  </apply>  </list>  </apply>  <ms> , </ms>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> 1 </ms>  </apply>  <ms> , </ms>  <ms> … </ms>  <ms> , </ms>  <apply>  <ci> RowBox </ci>  <list>  <apply>  <ci> SubscriptBox </ci>  <ms> b </ms>  <ms> q </ms>  </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>  </list>  </apply>  <ms> ) </ms>  </list>  </apply>  </list>  </apply>  </list>  </apply>  </list>  </apply>  <ms> /; </ms>  <apply>  <ci> RowBox </ci>  <list>  <ms> n </ms>  <ms> ∈ </ms>  <apply>  <ci> SuperscriptBox </ci>  <ms> ℕ </ms>  <ms> + </ms>  </apply>  </list>  </apply>  </list>  </apply>  <ci> TraditionalForm </ci>  </apply>  </annotation-xml>  </semantics>  </math> 
   
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  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z_", "\[Alpha]_"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n_"]], ",", SubscriptBox["a_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", "z_"]], "]"]]]], ")"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "+", "1"]], "]"]], " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "n"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["\[Alpha]", "+", "1"]], ",", SubscriptBox["aa", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["aa", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["\[Alpha]", "-", "n", "+", "1"]], ",", SubscriptBox["bb", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["bb", "q"]]], "}"]], ",", "z"]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]  |  
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   Date Added to functions.wolfram.com (modification date)
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