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http://functions.wolfram.com/07.27.13.0019.01
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(1 - z) z^2 Derivative[3][w][z] + (Subscript[b, 1] + Subscript[b, 2] + 1 -
(Subscript[a, 1] + Subscript[a, 2] + Subscript[a, 3] + 3) z) z
Derivative[2][w][z] + (Subscript[b, 1] Subscript[b, 2] -
(Subscript[a, 1] Subscript[a, 2] + Subscript[a, 2] Subscript[a, 3] +
Subscript[a, 1] Subscript[a, 3] + Subscript[a, 1] + Subscript[a, 2] +
Subscript[a, 3] + 1) z) Derivative[1][w][z] -
Subscript[a, 1] Subscript[a, 2] Subscript[a, 3] w[z] == 0 /;
w[z] == (Subscript[c, 1] HypergeometricPFQRegularized[
{Subscript[a, 1], 1 + Subscript[a, 1] - Subscript[b, 1],
1 + Subscript[a, 1] - Subscript[b, 2]},
{1 + Subscript[a, 1] - Subscript[a, 2], 1 + Subscript[a, 1] -
Subscript[a, 3]}, 1/z])/z^Subscript[a, 1] +
(Subscript[c, 2] HypergeometricPFQRegularized[{Subscript[a, 2],
1 + Subscript[a, 2] - Subscript[b, 1], 1 + Subscript[a, 2] -
Subscript[b, 2]}, {1 + Subscript[a, 2] - Subscript[a, 1],
1 + Subscript[a, 2] - Subscript[a, 3]}, 1/z])/z^Subscript[a, 2] +
(Subscript[c, 3] HypergeometricPFQRegularized[{Subscript[a, 3],
1 + Subscript[a, 3] - Subscript[b, 1], 1 + Subscript[a, 3] -
Subscript[b, 2]}, {1 + Subscript[a, 3] - Subscript[a, 1],
1 + Subscript[a, 3] - Subscript[a, 2]}, 1/z])/z^Subscript[a, 3] &&
!Element[Subscript[a, 1] - Subscript[a, 2], Integers] &&
!Element[Subscript[a, 1] - Subscript[a, 3], Integers] &&
!Element[Subscript[a, 2] - Subscript[a, 3], Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], " ", SuperscriptBox["z", "2"], " ", RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"], "+", "1", "-", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["a", "2"], "+", SubscriptBox["a", "3"], "+", "3"]], ")"]], " ", "z"]]]], ")"]], " ", "z", " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["b", "1"], " ", SubscriptBox["b", "2"]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["a", "1"], " ", SubscriptBox["a", "2"]]], "+", RowBox[List[SubscriptBox["a", "2"], " ", SubscriptBox["a", "3"]]], "+", RowBox[List[SubscriptBox["a", "1"], " ", SubscriptBox["a", "3"]]], "+", SubscriptBox["a", "1"], "+", SubscriptBox["a", "2"], "+", SubscriptBox["a", "3"], "+", "1"]], ")"]], " ", "z"]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "-", RowBox[List[SubscriptBox["a", "1"], " ", SubscriptBox["a", "2"], " ", SubscriptBox["a", "3"], " ", RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["z", RowBox[List["-", SubscriptBox["a", "1"]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"]]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", RowBox[List["-", SubscriptBox["a", "2"]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "2"], ",", RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"]]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "3"], " ", SuperscriptBox["z", RowBox[List["-", SubscriptBox["a", "3"]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "3"], ",", RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["a", "2"]]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]]]]]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"]]], "\[Element]", "Integers"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"]]], "\[Element]", "Integers"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"]]], "\[Element]", "Integers"]], "]"]]]], ")"]]]]]]
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</mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, 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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,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] | |
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