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http://functions.wolfram.com/07.28.20.0007.01
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Derivative[{0, 0, 0, 1}, {0, 0, 0}, 0][HypergeometricPFQ][
{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]},
{Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] ==
Sum[(Pochhammer[Subscript[a, 1], k] Pochhammer[Subscript[a, 2], k]
Pochhammer[Subscript[a, 3], k] Pochhammer[Subscript[a, 4], k]
PolyGamma[Subscript[a, 4] + k] z^k)/(Pochhammer[Subscript[b, 1], k]
Pochhammer[Subscript[b, 2], k] Pochhammer[Subscript[b, 3], k] k!),
{k, 0, Infinity}] - PolyGamma[Subscript[a, 4]]
HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3],
Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]},
z] /; Abs[z] < 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "0", ",", "0", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List["0", ",", "0", ",", "0"]], "}"]], ",", "0"]], "]"]], "[", "HypergeometricPFQ", "]"]], "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "1"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "3"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "4"], ",", "k"]], "]"]], RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "4"], "+", "k"]], "]"]], SuperscriptBox["z", "k"]]], ")"]], "/", " ", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "1"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "3"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]], ")"]]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", SubscriptBox["a", "4"], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]
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Date Added to functions.wolfram.com (modification date)
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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,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] | |
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