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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.20.0010.01

 Input Form

 Derivative[{0, \[Ellipsis], 0}, {n, 0, \[Ellipsis], 0}, 0][HypergeometricPFQ][ {Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == Sum[(Product[Pochhammer[Subscript[a, j], k], {j, 1, p}]/ (k! Product[Pochhammer[Subscript[b, j], k], {j, 2, q}])) D[1/Pochhammer[Subscript[b, 1], k], {Subscript[b, 1], n}] z^k, {k, 0, Infinity}] /; (Abs[z] < 1 && Element[n, Integers] && n > 0 && q == p - 1 && Abs[z] < 1) || q >= p

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "\[Ellipsis]", ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List["n", ",", "0", ",", "\[Ellipsis]", ",", "0"]], "}"]], ",", "0"]], "]"]], "[", "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"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "j"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "q"], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "j"], ",", "k"]], "]"]]]]]]], RowBox[List["D", "[", RowBox[List[FractionBox["1", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "1"], ",", "k"]], "]"]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "n"]], "}"]]]], "]"]], SuperscriptBox["z", "k"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["q", "\[Equal]", RowBox[List["p", "-", "1"]]]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]], "\[Or]", RowBox[List["q", "\[GreaterEqual]", "p"]]]]]]]]

 MathML Form

 p F q ( { 0 , , 0 } , { n , 0 , , 0 } , 0 ) ( a 1 , , a p TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], 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]] ; b 1 , , b q TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) k = 0 j = 1 p ( a j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] k ! j = 2 q ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] n 1 ( b 1 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "1"], ")"]], "k"], Pochhammer] b 1 n z k /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 n + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] q p - 1 "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 q p p F q ( { 0 , , 0 } , { n , 0 , , 0 } , 0 ) ( a 1 , , a p TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], 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]] ; b 1 , , b q TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) k = 0 j = 1 p ( a j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] k ! j = 2 q ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] n 1 ( b 1 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "1"], ")"]], "k"], Pochhammer] b 1 n z k /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 n + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] q p - 1 "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 q p [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["HypergeometricPFQ", TagBox[RowBox[List["(", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "\[Ellipsis]_", ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List["n", ",", "0", ",", "\[Ellipsis]_", ",", "0"]], "}"]], ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", 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_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "j"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", "n"]], "}"]]]]], FractionBox["1", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["bb", "1"], ",", "k"]], "]"]]]]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "q"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "j"], ",", "k"]], "]"]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["q", "\[Equal]", RowBox[List["p", "-", "1"]]]], "&&", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]], ")"]], "||", RowBox[List["q", "\[GreaterEqual]", "p"]]]]]]]]]]

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

 2001-10-29