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 HypergeometricPFQ

 http://functions.wolfram.com/07.28.17.0025.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] == Sum[(z^k/k!) (Product[Pochhammer[Subscript[a, j], k], {j, 1, 4}]/ Product[Pochhammer[Subscript[b, j], k], {j, 1, 3}]) HypergeometricPFQ[{1, (Subscript[a, 1] + k)/n, \[Ellipsis], (Subscript[a, 1] + k + n - 1)/n, \[Ellipsis], (Subscript[a, 4] + k)/n, \[Ellipsis], (Subscript[a, 4] + k + n - 1)/n}, {(k + 1)/n, \[Ellipsis], (k + n)/n, (Subscript[b, 1] + k)/n, \[Ellipsis], (Subscript[b, 1] + k + n - 1)/n, (Subscript[b, 2] + k)/n, \[Ellipsis], (Subscript[b, 2] + k + n - 1)/n, (Subscript[b, 3] + k)/n, \[Ellipsis], (Subscript[b, 3] + k + n - 1)/n}, z^n], {k, 0, n - 1}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[SuperscriptBox["z", "k"], RowBox[List["k", "!"]]], FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "4"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "j"], ",", "k"]], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "3"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "j"], ",", "k"]], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "4"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "4"], "+", "k", "+", "n", "-", "1"]], "n"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["k", "+", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["k", "+", "n"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "2"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "2"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "3"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "3"], "+", "k", "+", "n", "-", "1"]], "n"]]], "}"]], ",", SuperscriptBox["z", "n"]]], "]"]]]]]]]]]]

 MathML Form

 4 F 3 ( a 1 , a 2 , a 3 , a 4 ; b 1 , b 2 , b 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", 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]]]], 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]]]], 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 n - 1 z k j = 1 4 ( a j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] k ! j = 1 3 ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] 4 n + 1 F 4 n ( 1 , a 1 + k n , , a 1 + k + n - 1 n , , a 4 + k n , , a 4 + k + n - 1 n ; k + 1 n , , k + n n , k + b 1 n , , k + n + b 1 - 1 n , k + b 2 n , , k + n + b 2 - 1 n , k + b 3 n , , k + n + b 3 - 1 n ; z n ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List[RowBox[List["4", "n"]], "+", "1"]], TraditionalForm]], SubscriptBox["F", RowBox[List["4", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List["1", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "4"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", RowBox[List[FractionBox[RowBox[List[SubscriptBox["a", "4"], "+", "k", "+", "n", "-", "1"]], "n"], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "1"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "1"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "2"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "2"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "3"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "3"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", SuperscriptBox["z", "n"]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] 4 F 3 ( a 1 , a 2 , a 3 , a 4 ; b 1 , b 2 , b 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", 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]]]], 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]]]], 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 n - 1 z k j = 1 4 ( a j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] k ! j = 1 3 ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] 4 n + 1 F 4 n ( 1 , a 1 + k n , , a 1 + k + n - 1 n , , a 4 + k n , , a 4 + k + n - 1 n ; k + 1 n , , k + n n , k + b 1 n , , k + n + b 1 - 1 n , k + b 2 n , , k + n + b 2 - 1 n , k + b 3 n , , k + n + b 3 - 1 n ; z n ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List[RowBox[List["4", "n"]], "+", "1"]], TraditionalForm]], SubscriptBox["F", RowBox[List["4", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List["1", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "4"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", RowBox[List[FractionBox[RowBox[List[SubscriptBox["a", "4"], "+", "k", "+", "n", "-", "1"]], "n"], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "1"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "1"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "2"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "2"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "3"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "3"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", SuperscriptBox["z", "n"]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] [/itex]

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

 2001-10-29