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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.07.0001.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (1/(Gamma[Subscript[a, p]] Gamma[Subscript[b, q] - Subscript[a, p]])) Integrate[t^(Subscript[a, p] - 1) (1 - t)^(Subscript[b, q] - Subscript[a, p] - 1) HypergeometricPFQRegularized[ {Subscript[a, 1], \[Ellipsis], Subscript[a, p - 1]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q - 1]}, z t], {t, 0, 1}] /; (Re[Subscript[b, q]] > Re[Subscript[a, p]] > 0 && p <= q) || (Re[Subscript[b, q]] > Re[Subscript[a, p]] > 0 && p == q + 1 && Abs[Arg[1 - z]] < Pi)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "p"], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "q"], "-", SubscriptBox["a", "p"]]], "]"]]]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SuperscriptBox["t", RowBox[List[SubscriptBox["a", "p"], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], RowBox[List[SubscriptBox["b", "q"], "-", SubscriptBox["a", "p"], "-", "1"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["p", "-", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", RowBox[List["q", "-", "1"]]]]], "}"]], ",", RowBox[List["z", " ", "t"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", SubscriptBox["b", "q"], "]"]], ">", RowBox[List["Re", "[", SubscriptBox["a", "p"], "]"]], ">", "0"]], "\[And]", RowBox[List["p", "\[LessEqual]", "q"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", SubscriptBox["b", "q"], "]"]], ">", RowBox[List["Re", "[", SubscriptBox["a", "p"], "]"]], ">", "0"]], "\[And]", RowBox[List["p", "\[Equal]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["1", "-", "z"]], "]"]], "]"]], "<", "\[Pi]"]]]], ")"]]]]]]]]

 MathML Form

 p F ~ q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", "z"]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]] 1 Γ ( a p ) Γ ( b q - a p ) 0 1 t a p - 1 ( 1 - t ) b q - a p - 1 p - 1 F ~ q - 1 ( a 1 , , a p - 1 ; b 1 , , b q - 1 ; z t ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["p", "-", "1"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["q", "-", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["p", "-", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["q", "-", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", RowBox[List["z", " ", "t"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]] t /; Re ( b q ) > Re ( a p ) > 0 p q Re ( b q ) > Re ( a p ) > 0 p q + 1 "\[LeftBracketingBar]" arg ( 1 - z ) "\[RightBracketingBar]" < π p F ~ q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", "z"]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]] 1 Γ ( a p ) Γ ( b q - a p ) 0 1 t a p - 1 ( 1 - t ) b q - a p - 1 p - 1 F ~ q - 1 ( a 1 , , a p - 1 ; b 1 , , b q - 1 ; z t ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["p", "-", "1"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["q", "-", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["p", "-", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["q", "-", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", RowBox[List["z", " ", "t"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]] t /; Re ( b q ) > Re ( a p ) > 0 p q Re ( b q ) > Re ( a p ) > 0 p q + 1 "\[LeftBracketingBar]" arg ( 1 - z ) "\[RightBracketingBar]" < π [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[RowBox[List[SuperscriptBox["t", RowBox[List[SubscriptBox["aa", "p"], "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "t"]], ")"]], RowBox[List[SubscriptBox["bb", "q"], "-", SubscriptBox["aa", "p"], "-", "1"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["p", "-", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", RowBox[List["q", "-", "1"]]]]], "}"]], ",", RowBox[List["z", " ", "t"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "p"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "q"], "-", SubscriptBox["aa", "p"]]], "]"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", SubscriptBox["bb", "q"], "]"]], ">", RowBox[List["Re", "[", SubscriptBox["aa", "p"], "]"]], ">", "0"]], "&&", RowBox[List["p", "\[LessEqual]", "q"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", SubscriptBox["bb", "q"], "]"]], ">", RowBox[List["Re", "[", SubscriptBox["aa", "p"], "]"]], ">", "0"]], "&&", RowBox[List["p", "\[Equal]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["1", "-", "z"]], "]"]], "]"]], "<", "\[Pi]"]]]], ")"]]]]]]]]]]

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

 2001-10-29