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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.03.0031.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", 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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Sigma]", " "]]], FractionBox[RowBox[List["n", "!"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List[SubscriptBox["n", "k"], RowBox[List["(", RowBox[List[SubscriptBox["n", "k"], "-", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["j", "+", "1"]]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["j", "+", "1"]]]]], ")"]]]], "-", "1"]], ")"]]]]]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List[SubscriptBox["n", "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["j", "+", "1"]]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", RowBox[List["k", "+", "1"]]]], RowBox[List["q", "+", "1"]]], RowBox[List[SubscriptBox["n", "l"], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["l", "-", "1"]]], RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["j", "+", "1"]]]]], ")"]]]], "-", "1"]], ")"]]]]]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["\[Sigma]", "\[Equal]", RowBox[List["n", "+", "2"]]]], "\[And]", RowBox[List[SubscriptBox["a", "1"], "\[Equal]", RowBox[List["-", "n"]]]], "\[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"]]]], "\[And]", RowBox[List["\[Sigma]", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], SubscriptBox["n", "j"]]]]]]]]]]]

 MathML Form

 q + 1 F ~ q ( - n , 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[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[SubscriptBox["a", RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["n", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "-", SubscriptBox["n", RowBox[List["q", "+", "1"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] ( - 1 ) σ n ! j = 2 q + 1 Γ ( a j ) ( 1 2 k = 2 q + 1 n k ( n k - 1 ) j = 1 k - 1 ( a j - a j + 1 + n j + 1 ) ( j = 1 k - 1 ( a j - a j + 1 + n j + 1 ) - 1 ) + k = 2 q + 1 n k j = 1 k - 1 ( a j - a j + 1 + n j + 1 ) l = k + 1 q + 1 n l ( j = 1 l - 1 ( a j - a j + 1 + n j + 1 ) - 1 ) ) /; σ n + 2 a 1 - n n j + 2 j q + 1 σ j = 2 q + 1 n j Condition HypergeometricPFQ -1 n Subscript a 2 -1 σ n j 2 q 1 Gamma Subscript a j -1 1 2 k 2 q 1 Subscript n k Subscript n k -1 j 1 k -1 Subscript a j -1 Subscript a j 1 Subscript n j 1 j 1 k -1 Subscript a j -1 Subscript a j 1 Subscript n j 1 -1 k 2 q 1 Subscript n k j 1 k -1 Subscript a j -1 Subscript a j 1 Subscript n j 1 l k 1 q 1 Subscript n l j 1 l -1 Subscript a j -1 Subscript a j 1 Subscript n j 1 -1 σ n 2 Subscript a 1 -1 n Subscript n j SuperPlus 2 j q 1 σ j 2 q 1 Subscript n j [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n_"]], ",", 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"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Sigma]"], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List[SubscriptBox["n", "k"], " ", RowBox[List["(", RowBox[List[SubscriptBox["n", "k"], "-", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["j", "+", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["j", "+", "1"]]]]], ")"]]]], "-", "1"]], ")"]]]]]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List[SubscriptBox["n", "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["j", "+", "1"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", RowBox[List["k", "+", "1"]]]], RowBox[List["q", "+", "1"]]], RowBox[List[SubscriptBox["nn", "l"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["l", "-", "1"]]], RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", SubscriptBox["n", RowBox[List["j", "+", "1"]]]]], ")"]]]], "-", "1"]], ")"]]]]]]]]]]]]]]]], ")"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]], "/;", RowBox[List[RowBox[List["\[Sigma]", "\[Equal]", RowBox[List["n", "+", "2"]]]], "&&", RowBox[List[SubscriptBox["aa", "1"], "\[Equal]", RowBox[List["-", "n"]]]], "&&", RowBox[List[SubscriptBox["n", "j"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["n", "j"], ">", "0"]], "&&", RowBox[List["2", "\[LessEqual]", "j", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["\[Sigma]", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], SubscriptBox["n", "j"]]]]]]]]]]]]]

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

 2001-10-29