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 HypergeometricPFQ

 http://functions.wolfram.com/07.22.13.0013.01

 Input Form

 Derivative[3][w][z] + (-3 Log[r] - 3 Log[s] + Log[r] (1 + Subscript[b, 1] + Subscript[b, 2])) Derivative[2][w][z] + ((1 - a r^z) Log[r]^2 + 4 Log[r] Log[s] + 3 Log[s]^2 - Log[r] (Log[r] + 2 Log[s]) Subscript[b, 2] + Log[r] Subscript[b, 1] (-Log[r] - 2 Log[s] + Log[r] Subscript[b, 2])) Derivative[1][w][z] + ((-a) r^z Log[r]^2 (-Log[s] + Log[r] Subscript[a, 1]) - Log[s] (Log[r] + Log[s] - Log[r] Subscript[b, 1]) (Log[r] + Log[s] - Log[r] Subscript[b, 2])) w[z] == 0 /; w[z] == Subscript[c, 1] s^z HypergeometricPFQRegularized[{Subscript[a, 1]}, {Subscript[b, 1], Subscript[b, 2]}, a r^z] + Subscript[c, 2] s^z (MeijerG[{{1 - Subscript[a, 1]}, {}}, {{0, 1 - Subscript[b, 1]}, {1 - Subscript[b, 2]}}, a r^z] + MeijerG[{{1 - Subscript[a, 1]}, {}}, {{0, 1 - Subscript[b, 2]}, {1 - Subscript[b, 1]}}, a r^z]) + Subscript[c, 3] s^z MeijerG[{{1 - Subscript[a, 1]}, {}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2]}, {}}, (-a) r^z]

 Standard Form

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 MathML Form

 w ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) + ( ( b 1 + b 2 + 1 ) log ( r ) - 3 log ( r ) - 3 log ( s ) ) w ′′ ( z ) + ( ( 1 - a r z ) log 2 ( r ) + 4 log ( s ) log ( r ) - ( log ( r ) + 2 log ( s ) ) b 2 log ( r ) + b 1 ( b 2 log ( r ) - log ( r ) - 2 log ( s ) ) log ( r ) + 3 log 2 ( s ) ) w ( z ) + ( - a log 2 ( r ) ( log ( r ) a 1 - log ( s ) ) r z - log ( s ) ( - b 1 log ( r ) + log ( r ) + log ( s ) ) ( - b 2 log ( r ) + log ( r ) + log ( s ) ) ) w ( z ) 0 /; w ( z ) c 1 s z 1 F ~ 2 ( a 1 ; b 1 , b 2 ; a r z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List["a", " ", SuperscriptBox["r", "z"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + c 2 s z ( G 1 , 3 2 , 1 ( a r z 1 - a 1 0 , 1 - b 1 , 1 - b 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "3"]], RowBox[List["2", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["a", " ", SuperscriptBox["r", "z"]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] + G 1 , 3 2 , 1 ( a r z 1 - a 1 0 , 1 - b 2 , 1 - b 1 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "3"]], RowBox[List["2", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["a", " ", SuperscriptBox["r", "z"]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) + c 3 s z G 1 , 3 3 , 1 ( - a r z 1 - a 1 0 , 1 - b 1 , 1 - b 2 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "3"]], RowBox[List["3", ",", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["r", "z"]]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] Condition z 3 w z Subscript b 1 Subscript b 2 1 r -1 3 r -1 3 s z 2 w z 1 -1 a r z r 2 4 s r -1 r 2 s Subscript b 2 r Subscript b 1 Subscript b 2 r -1 r -1 2 s r 3 s 2 z w z -1 a r 2 r Subscript a 1 -1 s r z -1 s -1 Subscript b 1 r r s -1 Subscript b 2 r r s w z 0 w z Subscript c 1 s z HypergeometricPFQRegularized Subscript a 1 Subscript b 1 Subscript b 2 a r z Subscript c 2 s z MeijerG 1 -1 Subscript a 1 0 1 -1 Subscript b 1 1 -1 Subscript b 2 a r z MeijerG 1 -1 Subscript a 1 0 1 -1 Subscript b 2 1 -1 Subscript b 1 a r z Subscript c 3 s z MeijerG 1 -1 Subscript a 1 0 1 -1 Subscript b 1 1 -1 Subscript b 2 -1 a r z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["w", TagBox[RowBox[List["(", "3", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "z_", "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", RowBox[List["Log", "[", "r_", "]"]]]], "-", RowBox[List["3", " ", RowBox[List["Log", "[", "s_", "]"]]]], "+", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", SubscriptBox["b_", "1"], "+", SubscriptBox["b_", "2"]]], ")"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["a_", " ", SuperscriptBox["r_", "z_"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Log", "[", "r_", "]"]], "2"]]], "+", RowBox[List["4", " ", RowBox[List["Log", "[", "r_", "]"]], " ", RowBox[List["Log", "[", "s_", "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["Log", "[", "s_", "]"]], "2"]]], "-", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", "s_", "]"]]]]]], ")"]], " ", SubscriptBox["b_", "2"]]], "+", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], " ", SubscriptBox["b_", "1"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "r_", "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", "s_", "]"]]]], "+", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], " ", SubscriptBox["b_", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a_"]], " ", SuperscriptBox["r_", "z_"], " ", SuperscriptBox[RowBox[List["Log", "[", "r_", "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "s_", "]"]]]], "+", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], " ", SubscriptBox["a_", "1"]]]]], ")"]]]], "-", RowBox[List[RowBox[List["Log", "[", "s_", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], "+", RowBox[List["Log", "[", "s_", "]"]], "-", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], " ", SubscriptBox["b_", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], "+", RowBox[List["Log", "[", "s_", "]"]], "-", RowBox[List[RowBox[List["Log", "[", "r_", "]"]], " ", SubscriptBox["b_", "2"]]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["s", "z"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", SubscriptBox["a", "1"], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["s", "z"], " ", RowBox[List["(", RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["a", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", SubscriptBox["b", "2"]]], "}"]]]], "}"]], ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]], "+", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["a", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", SubscriptBox["b", "1"]]], "}"]]]], "}"]], ",", RowBox[List["a", " ", SuperscriptBox["r", "z"]]]]], "]"]]]], ")"]]]], "+", RowBox[List[SubscriptBox["c", "3"], " ", SuperscriptBox["s", "z"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", "-", SubscriptBox["a", "1"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["r", "z"]]]]], "]"]]]]]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02