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 HypergeometricPFQ

 http://functions.wolfram.com/07.25.17.0021.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2]}, z] == Sum[(z^k/k!) ((Pochhammer[Subscript[a, 1], k] Pochhammer[Subscript[a, 2], k])/(Pochhammer[Subscript[b, 1], k] Pochhammer[Subscript[b, 2], k])) HypergeometricPFQ[{1, (Subscript[a, 1] + k)/n, \[Ellipsis], (Subscript[a, 1] + k + n - 1)/n, (Subscript[a, 2] + k)/n, \[Ellipsis], (Subscript[a, 2] + 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}, z^n/n^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"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "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[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "1"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "1"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "2"], ",", "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"], ",", FractionBox[RowBox[List[SubscriptBox["a", "2"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "2"], "+", "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[SuperscriptBox["z", "n"], SuperscriptBox["n", "n"]]]], "]"]]]]]]]]]]

 MathML Form

 p F q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", "z"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] k = 0 n - 1 z k j = 1 p ( a j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] k ! j = 1 q ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] n p + 1 F n q + n ( 1 , k + a 1 n , , k + a 1 + n - 1 n , k + a 2 n , , k + a 2 + n - 1 n ; k + 1 n , , k + n n , k + b 1 n , , k + b 1 + n - 1 n , k + b 2 n , , k + b 2 + n - 1 n ; z n n n ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List[RowBox[List["n", " ", "p"]], "+", "1"]], TraditionalForm]], SubscriptBox["F", RowBox[List[RowBox[List["n", " ", "q"]], "+", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "1"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "1"], "+", "n", "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "2"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "2"], "+", "n", "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ";", 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", "+", SubscriptBox["b", "1"], "+", "n", "-", "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]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "2"], "+", "n", "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[FractionBox[SuperscriptBox["z", "n"], SuperscriptBox["n", "n"]], HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] p F q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", "z"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] k = 0 n - 1 z k j = 1 p ( a j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] k ! j = 1 q ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] n p + 1 F n q + n ( 1 , k + a 1 n , , k + a 1 + n - 1 n , k + a 2 n , , k + a 2 + n - 1 n ; k + 1 n , , k + n n , k + b 1 n , , k + b 1 + n - 1 n , k + b 2 n , , k + b 2 + n - 1 n ; z n n n ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List[RowBox[List["n", " ", "p"]], "+", "1"]], TraditionalForm]], SubscriptBox["F", RowBox[List[RowBox[List["n", " ", "q"]], "+", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "1"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "1"], "+", "n", "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "2"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "2"], "+", "n", "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ";", 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", "+", SubscriptBox["b", "1"], "+", "n", "-", "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]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "2"], "+", "n", "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[FractionBox[SuperscriptBox["z", "n"], SuperscriptBox["n", "n"]], HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], 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