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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.23.0001.01

 Input Form

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

 Standard Form

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

 MathML Form

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

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

 2001-10-29