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 HypergeometricPFQ

 http://functions.wolfram.com/07.29.02.0001.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4], Subscript[a, 5]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3], Subscript[b, 4]}, z] == Sum[(Pochhammer[Subscript[a, 1], k] Pochhammer[Subscript[a, 2], k] Pochhammer[Subscript[a, 3], k] Pochhammer[Subscript[a, 4], k] Pochhammer[Subscript[a, 5], k] z^k)/(Pochhammer[Subscript[b, 1], k] Pochhammer[Subscript[b, 2], k] Pochhammer[Subscript[b, 3], k] Pochhammer[Subscript[b, 4], k] k!), {k, 0, Infinity}] /; Abs[z] < 1 || (Abs[z] == 1 && Re[Sum[Subscript[b, j], {j, 1, 4}] - Sum[Subscript[a, j], {j, 1, 5}]] > 0)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"], ",", SubscriptBox["a", "5"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"], ",", SubscriptBox["b", "4"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "1"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "2"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "3"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "4"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "5"], ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"]]], ")"]], "/", " ", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "1"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "2"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "4"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[RowBox[List["Re", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "4"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "5"], SubscriptBox["a", "j"]]]]], "]"]], ">", "0"]]]], ")"]]]]]]]]

 MathML Form

 5 F 4 ( a 1 , a 2 , a 3 , a 4 , a 5 ; b 1 , b 2 , b 3 , b 4 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["5", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "5"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] k = 0 ( a 1 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], "k"], Pochhammer] ( a 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "2"], ")"]], "k"], Pochhammer] ( a 3 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "3"], ")"]], "k"], Pochhammer] ( a 4 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "4"], ")"]], "k"], Pochhammer] ( a 5 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "5"], ")"]], "k"], Pochhammer] z k ( b 1 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "1"], ")"]], "k"], Pochhammer] ( b 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "2"], ")"]], "k"], Pochhammer] ( b 3 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "3"], ")"]], "k"], Pochhammer] ( b 4 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "4"], ")"]], "k"], Pochhammer] k ! /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 "\[LeftBracketingBar]" z "\[RightBracketingBar]" 1 Re ( j = 1 4 b j - j = 1 5 a j ) > 0 Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript a 4 Subscript a 5 Subscript b 1 Subscript b 2 Subscript b 3 Subscript b 4 z k 0 Pochhammer Subscript a 1 k Pochhammer Subscript a 2 k Pochhammer Subscript a 3 k Pochhammer Subscript a 4 k Pochhammer Subscript a 5 k z k Pochhammer Subscript b 1 k Pochhammer Subscript b 2 k Pochhammer Subscript b 3 k Pochhammer Subscript b 4 k k -1 z 1 z 1 j 1 4 Subscript b j -1 j 1 5 Subscript a j 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", SubscriptBox["a_", "2"], ",", SubscriptBox["a_", "3"], ",", SubscriptBox["a_", "4"], ",", SubscriptBox["a_", "5"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", SubscriptBox["b_", "3"], ",", SubscriptBox["b_", "4"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["aa", "1"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["aa", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["aa", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["aa", "4"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["aa", "5"], ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["bb", "1"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["bb", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["bb", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["bb", "4"], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List["Re", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "4"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "5"], SubscriptBox["a", "j"]]]]], "]"]], ">", "0"]]]], ")"]]]]]]]]]]

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

 2001-10-29