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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.20.0016.01

 Input Form

 D[z^\[Alpha] HypergeometricPFQRegularized[{-(n/r), (-n + 1)/r, \[Ellipsis], (-n + r - 1)/r, Subscript[a, r + 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z^m], {z, n}] == m^(n - \[Alpha] - 1/2) (2 Pi)^((m - 1)/2) Gamma[\[Alpha] + 1] z^(\[Alpha] - n) HypergeometricPFQRegularized[{-(n/r), (-n + 1)/r, \[Ellipsis], (-n + r - 1)/r, (\[Alpha] + 1)/m, (\[Alpha] + 2)/m, \[Ellipsis], (\[Alpha] + m)/m, Subscript[a, r + 1], \[Ellipsis], Subscript[a, p]}, {(\[Alpha] - n + 1)/m, (\[Alpha] - n + 2)/m, \[Ellipsis], (\[Alpha] - n + m)/m, Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z^m] /; Element[r, Integers] && r > 0 && Element[m, Integers] && m > 0 && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["(", RowBox[List[SuperscriptBox["z", "\[Alpha]"], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n", "r"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "r"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "r", "-", "1"]], "r"], ",", SubscriptBox["a", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", SuperscriptBox["z", "m"]]], "]"]]]], ")"]]]], "\[Equal]", " ", RowBox[List[SuperscriptBox["m", RowBox[List["n", "-", "\[Alpha]", "-", FractionBox["1", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "\[Pi]"]], ")"]], FractionBox[RowBox[List["m", "-", "1"]], "2"]], RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "+", "1"]], "]"]], SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "n"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n", "r"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "r"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "r", "-", "1"]], "r"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "m"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "m"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["\[Alpha]", "+", "m"]], "m"], ",", SubscriptBox["a", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "1"]], "m"], ",", FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "2"]], "m"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "m"]], "m"], ",", SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", SuperscriptBox["z", "m"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["r", "\[Element]", "Integers"]], "\[And]", RowBox[List["r", ">", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 n ( z α p F ~ q ( - n r , - n + 1 r , , - n + r - 1 r , a r + 1 , , a p TagBox[TagBox[RowBox[List[RowBox[List["-", FractionBox["n", "r"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "r"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "r", "-", "1"]], "r"], ",", SubscriptBox["a", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; b 1 , , b q ; z m TagBox[SuperscriptBox["z", "m"], HypergeometricPFQ, Rule[Editable, True]] ) ) z n m n - α - 1 2 ( 2 π ) m - 1 2 Γ ( α + 1 ) z α - n p + m F ~ q + m ( - n r , - n + 1 r , , - n + r - 1 r , α + 1 m , α + 2 m , , α + m m , a r + 1 , , a p TagBox[TagBox[RowBox[List[RowBox[List["-", FractionBox["n", "r"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "r"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "r", "-", "1"]], "r"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "m"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "m"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["\[Alpha]", "+", "m"]], "m"], ",", SubscriptBox["a", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; α - n + 1 m , α - n + 2 m , , α - n + m m , b 1 , , b q ; z m TagBox[SuperscriptBox["z", "m"], HypergeometricPFQ, Rule[Editable, True]] ) /; r + m + n + FormBox RowBox RowBox RowBox D [ RowBox ( RowBox SuperscriptBox z α RowBox RowBox SubscriptBox ErrorBox p SubscriptBox OverscriptBox F ~ FormBox q TraditionalForm RowBox ( RowBox RowBox TagBox TagBox RowBox RowBox - FractionBox n r , FractionBox RowBox RowBox - n + 1 r , , FractionBox RowBox RowBox - n + r - 1 r , SubscriptBox a RowBox r + 1 , , SubscriptBox a p InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; SubscriptBox b 1 , , RowBox SubscriptBox b q ; TagBox SuperscriptBox z m HypergeometricPFQ Rule Editable ) ) , RowBox { z , n } ] RowBox SuperscriptBox m RowBox n - α - FractionBox 1 2 SuperscriptBox RowBox ( RowBox 2 π ) FractionBox RowBox m - 1 2 RowBox Γ ( RowBox α + 1 ) SuperscriptBox z RowBox α - n RowBox RowBox SubscriptBox RowBox p + m SubscriptBox OverscriptBox F ~ FormBox RowBox q + m TraditionalForm RowBox ( RowBox RowBox TagBox TagBox RowBox RowBox - FractionBox n r , FractionBox RowBox RowBox - n + 1 r , , FractionBox RowBox RowBox - n + r - 1 r , FractionBox RowBox α + 1 m , FractionBox RowBox α + 2 m , , FractionBox RowBox α + m m , SubscriptBox a RowBox r + 1 , , SubscriptBox a p InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; FractionBox RowBox α - n + 1 m , FractionBox RowBox α - n + 2 m , , FractionBox RowBox α - n + m m , SubscriptBox b 1 , , RowBox SubscriptBox b q ; TagBox SuperscriptBox z m HypergeometricPFQ Rule Editable ) /; RowBox RowBox r SuperscriptBox + RowBox m SuperscriptBox + RowBox n SuperscriptBox + TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["z_", "\[Alpha]_"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n_", "r_"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n_"]], "+", "1"]], "r_"], ",", "\[Ellipsis]_", ",", FractionBox[RowBox[List[RowBox[List["-", "n_"]], "+", "r_", "-", "1"]], "r_"], ",", SubscriptBox["a_", RowBox[List["r_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", SuperscriptBox["z_", "m_"]]], "]"]]]], ")"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["m", RowBox[List["n", "-", "\[Alpha]", "-", FractionBox["1", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]"]], ")"]], FractionBox[RowBox[List["m", "-", "1"]], "2"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", "+", "1"]], "]"]], " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "n"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["n", "r"]]], ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "1"]], "r"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["-", "n"]], "+", "r", "-", "1"]], "r"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "m"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "m"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["\[Alpha]", "+", "m"]], "m"], ",", SubscriptBox["a", RowBox[List["r", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["aa", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "1"]], "m"], ",", FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "2"]], "m"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["\[Alpha]", "-", "n", "+", "m"]], "m"], ",", SubscriptBox["bb", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["bb", "q"]]], "}"]], ",", SuperscriptBox["z", "m"]]], "]"]]]], "/;", RowBox[List[RowBox[List["r", "\[Element]", "Integers"]], "&&", RowBox[List["r", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2001-10-29