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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.03.0151.01

 Input Form

 HypergeometricPFQRegularized[{-n, 1, 3/2, 3/2, 3/2, f, g, 1 - f, 1 - g}, {1/2, 1/2, 1/2, n + 2, 1 + f, 2 - f, 1 + g, 2 - g}, 1] == ((n! Sin[f Pi] Sin[g Pi])/((-1 + f) f (f - g) (-1 + g) g (-1 + f + g) Pi^(7/2))) (((2 g - 1)^2 f (f - 1))/(Pochhammer[g + 1, n] Pochhammer[2 - g, n]) - ((2 f - 1)^2 g (g - 1))/ (Pochhammer[f + 1, n] Pochhammer[2 - f, n])) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", "1", ",", FractionBox["3", "2"], ",", FractionBox["3", "2"], ",", FractionBox["3", "2"], ",", "f", ",", "g", ",", RowBox[List["1", "-", "f"]], ",", RowBox[List["1", "-", "g"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["n", "+", "2"]], ",", RowBox[List["1", "+", "f"]], ",", RowBox[List["2", "-", "f"]], ",", RowBox[List["1", "+", "g"]], ",", RowBox[List["2", "-", "g"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["Sin", "[", RowBox[List["f", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["g", " ", "\[Pi]"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "f"]], ")"]], " ", "f", " ", RowBox[List["(", RowBox[List["f", "-", "g"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "g"]], ")"]], " ", "g", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "f", "+", "g"]], ")"]], " ", SuperscriptBox["\[Pi]", RowBox[List["7", "/", "2"]]]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", "g"]], "-", "1"]], ")"]], "2"], "f", RowBox[List["(", RowBox[List["f", "-", "1"]], ")"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["g", "+", "1"]], ",", "n"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "g"]], ",", "n"]], "]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", "f"]], "-", "1"]], ")"]], "2"], "g", RowBox[List["(", RowBox[List["g", "-", "1"]], ")"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["f", "+", "1"]], ",", "n"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "f"]], ",", "n"]], "]"]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 9 F ~ 8 ( - n , 1 , 3 2 , 3 2 , 3 2 , f , g , 1 - f , 1 - g ; 1 2 , 1 2 , 1 2 , n + 2 , f + 1 , 2 - f , g + 1 , 2 - g ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["9", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["8", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "n"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["f", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["g", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "f"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "g"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", "2"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["f", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", "f"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["g", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["2", "-", "g"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] n ! sin ( f π ) sin ( g π ) ( f - 1 ) f ( f - g ) ( g - 1 ) g ( f + g - 1 ) π 7 / 2 ( ( 2 g - 1 ) 2 f ( f - 1 ) ( g + 1 ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["g", "+", "1"]], ")"]], "n"], Pochhammer] ( 2 - g ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", "g"]], ")"]], "n"], Pochhammer] - ( 2 f - 1 ) 2 g ( g - 1 ) ( f + 1 ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["f", "+", "1"]], ")"]], "n"], Pochhammer] ( 2 - f ) n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", "f"]], ")"]], "n"], Pochhammer] ) /; n Condition HypergeometricPFQRegularized -1 n 1 3 2 3 2 3 2 f g 1 -1 f 1 -1 g 1 2 1 2 1 2 n 2 f 1 2 -1 f g 1 2 -1 g 1 n f g f -1 f f -1 g g -1 g f g -1 7 2 -1 2 g -1 2 f f -1 Pochhammer g 1 n Pochhammer 2 -1 g n -1 -1 2 f -1 2 g g -1 Pochhammer f 1 n Pochhammer 2 -1 f n -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n_"]], ",", "1", ",", FractionBox["3", "2"], ",", FractionBox["3", "2"], ",", FractionBox["3", "2"], ",", "f_", ",", "g_", ",", RowBox[List["1", "-", "f_"]], ",", RowBox[List["1", "-", "g_"]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["n_", "+", "2"]], ",", RowBox[List["1", "+", "f_"]], ",", RowBox[List["2", "-", "f_"]], ",", RowBox[List["1", "+", "g_"]], ",", RowBox[List["2", "-", "g_"]]]], "}"]], ",", "1"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["Sin", "[", RowBox[List["f", " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["g", " ", "\[Pi]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "g"]], "-", "1"]], ")"]], "2"], " ", "f", " ", RowBox[List["(", RowBox[List["f", "-", "1"]], ")"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["g", "+", "1"]], ",", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "g"]], ",", "n"]], "]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "f"]], "-", "1"]], ")"]], "2"], " ", "g", " ", RowBox[List["(", RowBox[List["g", "-", "1"]], ")"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["f", "+", "1"]], ",", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "f"]], ",", "n"]], "]"]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "f"]], ")"]], " ", "f", " ", RowBox[List["(", RowBox[List["f", "-", "g"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "g"]], ")"]], " ", "g", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "f", "+", "g"]], ")"]], " ", SuperscriptBox["\[Pi]", RowBox[List["7", "/", "2"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29