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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.03.0038.01

 Input Form

 HypergeometricPFQRegularized[{-n, a/(q + 1), (a + 1)/(q + 1), \[Ellipsis], (a + q)/(q + 1), b/q, (b + 1)/q, \[Ellipsis], (b + q - 1)/q}, {(a + 1)/q, (a + 2)/q, \[Ellipsis], (a + q)/q, (b - 1)/(q + 1), b/(q + 1), \[Ellipsis], (b - 1 + q)/(q + 1)}, 1] == (2 Pi)^(1/2 - q) q^(1/2 + a) (1 + q)^(b - 3/2) Sum[(Pochhammer[-n, k] Pochhammer[a, q k + k] Pochhammer[b, q k])/(k! Gamma[a + 1 + q k] Gamma[b - 1 + q k + k]), {k, 0, n}] /; Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 2 q + 2 F ~ 2 q + 1 ( - n , a q + 1 , a + 1 q + 1 , , a + q q + 1 , b q , b + 1 q , , b + q - 1 q ; a + 1 q , a + 2 q , , a + q q , b - 1 q + 1 , b q + 1 , , b - 1 + q q + 1 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List[RowBox[List["2", "q"]], "+", "2"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List[RowBox[List["2", "q"]], "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["a", RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "1"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "q"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "+", "1"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["b", "+", "q", "-", "1"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["a", "+", "1"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "2"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "q"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "-", "1"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["b", RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "-", "1", "+", "q"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] ( 2 π ) 1 2 - q q a + 1 2 ( q + 1 ) b - 3 2 k = 0 n ( - n ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], "k"], Pochhammer] ( a ) q k + k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], RowBox[List[RowBox[List["q", " ", "k"]], "+", "k"]]], Pochhammer] ( b ) q k TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], RowBox[List["q", " ", "k"]]], Pochhammer] k ! Γ ( a + k q + 1 ) Γ ( b + k + k q - 1 ) /; n + Condition HypergeometricPFQ -1 n a q 1 -1 a 1 q 1 -1 2 1 2 -1 q q a 1 2 q 1 b -1 3 2 k 0 n Pochhammer -1 n k Pochhammer a q k k Pochhammer b q k k Gamma a k q 1 Gamma b k k q -1 -1 n SuperPlus [/itex]

 Rule Form

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

 2001-10-29