html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.27.13.0004.01

 Input Form

 (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], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] + Subscript[c, 2] z^(1 - Subscript[b, 1]) HypergeometricPFQRegularized[ {1 + Subscript[a, 1] - Subscript[b, 1], 1 + Subscript[a, 2] - Subscript[b, 1], 1 + Subscript[a, 3] - Subscript[b, 1]}, {2 - Subscript[b, 1], 1 - Subscript[b, 1] + Subscript[b, 2]}, z] + Subscript[c, 3] z^(1 - Subscript[b, 2]) HypergeometricPFQRegularized[ {1 + Subscript[a, 1] - Subscript[b, 2], 1 + Subscript[a, 2] - Subscript[b, 2], 1 + Subscript[a, 3] - Subscript[b, 2]}, {2 - Subscript[b, 2], 1 + Subscript[b, 1] - Subscript[b, 2]}, z] && !Element[Subscript[b, 1], Integers] && !Element[Subscript[b, 2], Integers] && !Element[Subscript[b, 1] - Subscript[b, 2], Integers]

 Standard Form

 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["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", RowBox[List["1", "-", SubscriptBox["b", "1"]]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", SubscriptBox["b", "1"]]], ",", RowBox[List["1", "-", SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"]]]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "3"], " ", SuperscriptBox["z", RowBox[List["1", "-", SubscriptBox["b", "2"]]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", SubscriptBox["b", "2"]]], ",", RowBox[List["1", "+", SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"]]]]], "}"]], ",", "z"]], "]"]]]]]]]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[SubscriptBox["b", "1"], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[SubscriptBox["b", "2"], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"]]], ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 ( 1 - z ) w ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) z 2 + ( - ( a 1 + a 2 + a 3 + 3 ) z + b 1 + b 2 + 1 ) z w ′′ ( z ) + ( b 1 b 2 - ( a 2 a 1 + a 3 a 1 + a 1 + a 2 + a 2 a 3 + a 3 + 1 ) z ) w ( z ) - a 1 a 2 a 3 w ( z ) 0 /; w ( z ) c 1 3 F ~ 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], 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["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + c 2 z 1 - b 1 3 F ~ 2 ( a 1 - b 1 + 1 , a 2 - b 1 + 1 , a 3 - b 1 + 1 ; 2 - b 1 , 1 - b 1 + b 2 ; z ) 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", "1"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", SubscriptBox["b", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], HypergeometricPFQ] + c 3 z 1 - b 2 3 F ~ 2 ( a 1 - b 2 + 1 , a 2 - b 2 + 1 , a 3 - b 2 + 1 TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]] ; 2 - b 2 , b 1 - b 2 + 1 TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", SubscriptBox["b", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]] ; z TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]] ) b 1 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] b 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] b 1 - b 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] ( 1 - z ) w ( 3 ) TagBox[RowBox[List["(", "3", ")"]], Derivative] ( z ) z 2 + ( - ( a 1 + a 2 + a 3 + 3 ) z + b 1 + b 2 + 1 ) z w ′′ ( z ) + ( b 1 b 2 - ( a 2 a 1 + a 3 a 1 + a 1 + a 2 + a 2 a 3 + a 3 + 1 ) z ) w ( z ) - a 1 a 2 a 3 w ( z ) 0 /; w ( z ) c 1 3 F ~ 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "3"], SubscriptBox[OverscriptBox["F", "~"], "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], 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["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] + c 2 z 1 - b 1 3 F ~ 2 ( a 1 - b 1 + 1 , a 2 - b 1 + 1 , a 3 - b 1 + 1 ; 2 - b 1 , 1 - b 1 + b 2 ; z ) 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", "1"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", SubscriptBox["b", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], HypergeometricPFQ] + c 3 z 1 - b 2 3 F ~ 2 ( a 1 - b 2 + 1 , a 2 - b 2 + 1 , a 3 - b 2 + 1 TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]] ; 2 - b 2 , b 1 - b 2 + 1 TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", SubscriptBox["b", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]] ; z TagBox["z", HypergeometricPFQRegularized, Rule[Editable, True]] ) b 1 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] b 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] b 1 - b 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] [/itex]

 Rule Form

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

 2007-05-02