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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.20.0021.01

 Input Form

 D[HypergeometricPFQ[{a, Subscript[a, 2], \[Ellipsis], Subscript[a, p]}, {a + 1, Subscript[b, 2], \[Ellipsis], Subscript[b, q]}, z], {a, n}] == (((-1)^(n - 1) n! z)/(1 + a)^(n + 1)) (Product[Subscript[a, j], {j, 2, p}]/ Product[Subscript[b, j], {j, 2, q}]) HypergeometricPFQ[ {Subscript[c, 1], \[Ellipsis], Subscript[c, n + 1], Subscript[a, 2] + 1, \[Ellipsis], Subscript[a, p] + 1}, {Subscript[c, 1] + 1, \[Ellipsis], Subscript[c, n + 1] + 1, Subscript[b, 2] + 1, \[Ellipsis], Subscript[b, q] + 1}, z] /; Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == a + 1 && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "n"]], "}"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "+", "1"]], ",", SubscriptBox["b", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], RowBox[List["n", "!"]], "z"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "a"]], ")"]], RowBox[List["n", "+", "1"]]]], FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "p"], SubscriptBox["a", "j"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "q"], " ", SubscriptBox["b", "j"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["c", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["c", RowBox[List["n", "+", "1"]]], ",", RowBox[List[SubscriptBox["a", "2"], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], "+", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["c", "1"], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["c", RowBox[List["n", "+", "1"]]], "+", "1"]], ",", RowBox[List[SubscriptBox["b", "2"], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], "+", "1"]]]], "}"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "1"], "\[Equal]", SubscriptBox["c", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["c", RowBox[List["n", "+", "1"]]], "\[Equal]", RowBox[List["a", "+", "1"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

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

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

 2003-08-21