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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.03.0025.01

 Input Form

 HypergeometricPFQ[{a, Subscript[a, 2], \[Ellipsis], Subscript[a, q + 1]}, {Subscript[a, 2] + Subscript[n, 2], \[Ellipsis], Subscript[a, q + 1] + Subscript[n, q + 1]}, 1] == Gamma[1 - a] Sum[(Gamma[Subscript[a, k] + Subscript[n, k]]/ ((Subscript[n, k] - 1)! Gamma[Subscript[a, k] - a + 1])) Sum[((Pochhammer[1 - Subscript[n, k], j] Pochhammer[Subscript[a, k], j])/ (j! Pochhammer[Subscript[a, k] - a + 1, j])) Product[If[l == k, 1, (Pochhammer[Subscript[a, l], Subscript[n, l]] Pochhammer[1 - Subscript[a, l] + Subscript[a, k] - Subscript[n, l], j])/(Pochhammer[Subscript[a, l] - Subscript[a, k], Subscript[n, l]] Pochhammer[ 1 - Subscript[a, l] + Subscript[a, k], j])], {l, 2, q + 1}], {j, 0, Subscript[n, k] - 1}], {k, 2, q + 1}] /; Re[a] < Sum[Subscript[n, j], {j, 2, q + 1}] && Element[Subscript[n, j], Integers] && Subscript[n, j] > 0 && 2 <= j <= q + 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["n", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["q", "+", "1"]]]]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "a"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "+", SubscriptBox["n", "k"]]], "]"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["n", "k"], "-", "1"]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "-", "a", "+", "1"]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[SubscriptBox["n", "k"], "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["n", "k"]]], ",", "j"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "k"], ",", "j"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "-", "a", "+", "1"]], ",", "j"]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["l", "\[Equal]", "k"]], ",", "1", ",", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "l"], ",", SubscriptBox["n", "l"]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", "l"], "+", SubscriptBox["a", "k"], "-", SubscriptBox["n", "l"]]], ",", "j"]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[SubscriptBox["a", "l"], "-", SubscriptBox["a", "k"]]], ",", SubscriptBox["n", "l"]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", "l"], "+", SubscriptBox["a", "k"]]], ",", "j"]], "]"]]]], ")"]]]]]], "]"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "a", "]"]], "<", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], SubscriptBox["n", "j"]]]]], "\[And]", RowBox[List[SubscriptBox["n", "j"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "j"], ">", "0"]], "\[And]", RowBox[List["2", "\[LessEqual]", "j", "\[LessEqual]", RowBox[List["q", "+", "1"]]]]]]]]]]

 MathML Form

 q + 1 F q ( a , a 2 , , a q + 1 ; a 2 + n 2 , , a q + 1 + n q + 1 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List["a", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], ";", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["n", "2"]]]]], ",", "\[Ellipsis]", ",", RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["q", "+", "1"]]]]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] Γ ( 1 - a ) k = 2 q + 1 Γ ( a k + n k ) ( n k - 1 ) ! Γ ( a k - a + 1 ) j = 0 n k - 1 ( 1 - n k ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["n", "k"]]], ")"]], "j"], Pochhammer] ( a k ) j TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "k"], ")"]], "j"], Pochhammer] j ! ( a k - a + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", "a", "+", "1"]], ")"]], "j"], Pochhammer] l = 2 l k q + 1 ( a l ) n l TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "l"], ")"]], SubscriptBox["n", "l"]], Pochhammer] ( a k - a l - n l + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", "l"], "-", SubscriptBox["n", "l"], "+", "1"]], ")"]], "j"], Pochhammer] ( a l - a k ) n l TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "l"], "-", SubscriptBox["a", "k"]]], ")"]], SubscriptBox["n", "l"]], Pochhammer] ( a k - a l + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", "l"], "+", "1"]], ")"]], "j"], Pochhammer] /; Re ( a ) < j = 2 q + 1 n j n j + 2 j q + 1 q + 1 F q ( a , a 2 , , a q + 1 ; a 2 + n 2 , , a q + 1 + n q + 1 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List["a", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], ";", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["n", "2"]]]]], ",", "\[Ellipsis]", ",", RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["q", "+", "1"]]]]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] Γ ( 1 - a ) k = 2 q + 1 Γ ( a k + n k ) ( n k - 1 ) ! Γ ( a k - a + 1 ) j = 0 n k - 1 ( 1 - n k ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["n", "k"]]], ")"]], "j"], Pochhammer] ( a k ) j TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "k"], ")"]], "j"], Pochhammer] j ! ( a k - a + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", "a", "+", "1"]], ")"]], "j"], Pochhammer] l = 2 l k q + 1 ( a l ) n l TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "l"], ")"]], SubscriptBox["n", "l"]], Pochhammer] ( a k - a l - n l + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", "l"], "-", SubscriptBox["n", "l"], "+", "1"]], ")"]], "j"], Pochhammer] ( a l - a k ) n l TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "l"], "-", SubscriptBox["a", "k"]]], ")"]], SubscriptBox["n", "l"]], Pochhammer] ( a k - a l + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "-", SubscriptBox["a", "l"], "+", "1"]], ")"]], "j"], Pochhammer] /; Re ( a ) < j = 2 q + 1 n j n j + 2 j q + 1 [/itex]

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

 2001-10-29