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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.10.0002.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == (1/Product[Gamma[Subscript[b, k]], {k, 1, q}]) (1 + (z Product[Subscript[a, k], {k, 1, p}])/Product[Subscript[b, k], {k, 1, q}]/(1 + ContinueFraction[ {-((z Product[Subscript[a, j] + k, {j, 1, p}])/ ((k + 1) Product[Subscript[b, j] + k, {j, 1, q}])), 1 + (z Product[Subscript[a, j] + k, {j, 1, p}])/ ((k + 1) Product[Subscript[b, j] + k, {j, 1, q}])}, {k, 1, Infinity}]))

 Standard Form

 Cell[BoxData[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[RowBox[List[" ", "1"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List["z", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["a", "k"]]]]], RowBox[List[" ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], SubscriptBox["b", "k"]]]]]], "/", RowBox[List["(", RowBox[List["1", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["z", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "+", "k"]], ")"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["(", RowBox[List[SubscriptBox["b", "j"], "+", "k"]], ")"]]]]]]]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["z", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "+", "k"]], ")"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["(", RowBox[List[SubscriptBox["b", "j"], "+", "k"]], ")"]]]]]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 p F ~ q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "p"], 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 k = 1 q Γ ( b k ) ( 1 + ( z k = 1 p a k ) / ( ( k = 1 q b k ) ( 1 + Κ k ( - z j = 1 p ( k + a j ) ( k + 1 ) j = 1 q ( k + b j ) , z j = 1 p ( k + a j ) ( k + 1 ) j = 1 q ( k + b j ) + 1 ) 1 ) ) ) p F ~ q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "p"], 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 k = 1 q Γ ( b k ) ( 1 + ( z k = 1 p a k ) / ( ( k = 1 q b k ) ( 1 + Κ k ( - z j = 1 p ( k + a j ) ( k + 1 ) j = 1 q ( k + b j ) , z j = 1 p ( k + a j ) ( k + 1 ) j = 1 q ( k + b j ) + 1 ) 1 ) ) ) [/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]", FractionBox[RowBox[List["1", "+", FractionBox[RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["a", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], SubscriptBox["b", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "+", "k"]], ")"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["(", RowBox[List[SubscriptBox["b", "j"], "+", "k"]], ")"]]]]]]]]], ",", RowBox[List["1", "+", FractionBox[RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "+", "k"]], ")"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], RowBox[List["(", RowBox[List[SubscriptBox["b", "j"], "+", "k"]], ")"]]]]]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]], ")"]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]]]]]]]

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

 2001-10-29