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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.03.0007.01

 Input Form

 HypergeometricPFQRegularized[{1, Subscript[a, 2], \[Ellipsis], Subscript[a, p]}, {n + 1, Subscript[b, 2], \[Ellipsis], Subscript[b, q]}, z] == ((-1)^((p - 1) n)/z^n) (1/Product[Pochhammer[1 - Subscript[a, j], n], {j, 2, p}]) (HypergeometricPFQRegularized[{Subscript[a, 2] - n, \[Ellipsis], Subscript[a, p] - n}, {Subscript[b, 2] - n, \[Ellipsis], Subscript[b, q] - n}, z] - (1/Product[Gamma[Subscript[b, j] - n], {j, 2, q}]) Sum[(Product[Pochhammer[Subscript[a, j] - n, k], {j, 2, p}]/ Product[Pochhammer[Subscript[b, j] - n, k], {j, 2, q}]) (z^k/k!), {k, 0, n - 1}])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n", "+", "1"]], ",", SubscriptBox["b", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["p", "-", "1"]], ")"]], "n"]]], SuperscriptBox["z", "n"]], FractionBox["1", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "p"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", "j"]]], ",", "n"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "-", "n"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], "-", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["b", "2"], "-", "n"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], "-", "n"]]]], "}"]], ",", "z"]], "]"]], "-", RowBox[List[FractionBox["1", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", "n"]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "p"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", "n"]], ",", "k"]], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "q"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[SubscriptBox["b", "j"], "-", "n"]], ",", "k"]], "]"]]]]], FractionBox[SuperscriptBox["z", "k"], RowBox[List["k", "!"]]]]]]]]]]], ")"]]]]]]]]

 MathML Form

 p F ~ q ( 1 , a 2 , , a p ; n + 1 , b 2 , , b q ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Ellipsis]", ",", SubscriptBox["a", "p"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Ellipsis]", ",", SubscriptBox["b", "q"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] ( - 1 ) ( p - 1 ) n z n j = 2 p ( 1 - a j ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["a", "j"]]], ")"]], "n"], Pochhammer] ( p - 1 F ~ q - 1 ( a 2 - n , , a p - n TagBox[TagBox[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "2"], "-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], "-", "n"]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; b 2 - n , , b q - n TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "2"], "-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], "-", "n"]]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) - 1 j = 2 q Γ ( b j - n ) k = 0 n - 1 ( j = 2 p ( a j - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "j"], "-", "n"]], ")"]], "k"], Pochhammer] ) z k ( j = 2 q ( b j - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "j"], "-", "n"]], ")"]], "k"], Pochhammer] ) k ! ) FormBox RowBox TagBox RowBox RowBox SubscriptBox FormBox p TraditionalForm SubscriptBox OverscriptBox F ~ FormBox q TraditionalForm RowBox ( RowBox TagBox TagBox TagBox TagBox RowBox TagBox 1 HypergeometricPFQ Rule Editable , TagBox SubscriptBox a 2 HypergeometricPFQ Rule Editable , TagBox ErrorBox RowBox , SubscriptBox a p HypergeometricPFQ Rule Editable Function Null HoldComplete SlotSequence 1 HoldAllComplete HypergeometricPFQ Rule Editable Function Null HoldComplete SlotSequence 1 HoldAllComplete HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox RowBox n + 1 HypergeometricPFQ Rule Editable , TagBox SubscriptBox b 2 HypergeometricPFQ Rule Editable , TagBox RowBox , SubscriptBox b q HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox z HypergeometricPFQ Rule Editable ) Function Null HoldComplete HypergeometricPFQ Slot 1 Slot 2 Slot 3 HoldAllComplete RowBox FractionBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox RowBox ( RowBox p - 1 ) n RowBox SuperscriptBox z n RowBox UnderoverscriptBox RowBox j = 2 p TagBox SubscriptBox RowBox ( RowBox 1 - SubscriptBox a j ) n Pochhammer RowBox ( RowBox RowBox RowBox SubscriptBox RowBox FormBox p TraditionalForm - 1 SubscriptBox OverscriptBox F ~ RowBox q - 1 RowBox ( RowBox TagBox TagBox TagBox TagBox RowBox TagBox RowBox SubscriptBox a 2 - n HypergeometricPFQ Rule Editable , TagBox RowBox , RowBox SubscriptBox a p - n HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox RowBox SubscriptBox b 2 - n HypergeometricPFQ Rule Editable , TagBox RowBox , RowBox SubscriptBox b q - n HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox z HypergeometricPFQ Rule Editable ) - RowBox FractionBox 1 RowBox UnderoverscriptBox RowBox j = 2 q RowBox Γ ( RowBox SubscriptBox b j - n ) RowBox UnderoverscriptBox RowBox k = 0 RowBox n - 1 FractionBox RowBox RowBox ( RowBox UnderoverscriptBox RowBox j = 2 p TagBox SubscriptBox RowBox ( RowBox SubscriptBox a j - n ) k Pochhammer ) SuperscriptBox z k RowBox RowBox ( RowBox UnderoverscriptBox RowBox j = 2 q TagBox SubscriptBox RowBox ( RowBox SubscriptBox b j - n ) k Pochhammer ) RowBox k ! ) TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["a_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["n_", "+", "1"]], ",", SubscriptBox["b_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["p", "-", "1"]], ")"]], " ", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["aa", "2"], "-", "n"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["aa", "p"], "-", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["bb", "2"], "-", "n"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["bb", "q"], "-", "n"]]]], "}"]], ",", "z"]], "]"]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "p"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", "n"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "q"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[SubscriptBox["b", "j"], "-", "n"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["k", "!"]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", "n"]], "]"]]]]]]], ")"]]]], RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], "p"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", "j"]]], ",", "n"]], "]"]]]]]]]]]]]

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

 2001-10-29