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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.03.0011.01

 Input Form

 HypergeometricPFQRegularized[{1, (Subscript[a, 1] + m)/n, (Subscript[a, 1] + m + 1)/n, \[Ellipsis], (Subscript[a, 1] + m + n - 1)/ n, \[Ellipsis], (Subscript[a, p] + m)/n, (Subscript[a, p] + m + 1)/n, \[Ellipsis], (Subscript[a, p] + m + n - 1)/n}, {(m + 1)/n, (m + 2)/n, \[Ellipsis], (m + n)/n, (Subscript[b, 1] + m)/n, (Subscript[b, 1] + m + 1)/n, \[Ellipsis], (Subscript[b, 1] + m + n - 1)/ n, \[Ellipsis], (Subscript[b, q] + m)/n, (Subscript[b, q] + m + 1)/n, \[Ellipsis], (Subscript[b, q] + m + n - 1)/n}, z] == (((2 Pi)^(((1 - n)/2) (q + 1))/Product[Pochhammer[Subscript[a, j], m], {j, 1, p}]) n^\[Eta] Sum[Exp[-((2 Pi I k m)/n)] HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, n^(q - p + 1) Exp[(2 Pi I k)/n] z^(1/n)], {k, 0, n - 1}])/z^(m/n) /; Element[m, Integers] && m > 0 && Element[n, Integers] && n > 0 && \[Eta] == Sum[Subscript[b, k], {k, 1, q}] + m p - (q + 1)/2

 Standard Form

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 MathML Form

 n p + 1 F ~ n q + n ( 1 , a 1 + m n , a 1 + m + 1 n , , a 1 + m + n - 1 n , , a p + m n , a p + m + 1 n , , a p + m + n - 1 n TagBox[TagBox[TagBox[TagBox[RowBox[List["1", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "m"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "m", "+", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "m", "+", "n", "-", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "m"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "m", "+", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "m", "+", "n", "-", "1"]], "n"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; m + 1 n , m + 2 n , , m + n n , b 1 + m n , b 1 + m + 1 n , , b 1 + m + n - 1 n , , b q + m n , b q + m + 1 n , , b q + m + n - 1 n TagBox[TagBox[TagBox[TagBox[RowBox[List[FractionBox[RowBox[List["m", "+", "1"]], "n"], ",", FractionBox[RowBox[List["m", "+", "2"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["m", "+", "n"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "m"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "m", "+", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "m", "+", "n", "-", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "m"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "m", "+", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "m", "+", "n", "-", "1"]], "n"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) ( 2 π ) 1 - n 2 ( q + 1 ) n η j = 1 p ( a j ) m TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "m"], Pochhammer] z - m n k = 0 n - 1 exp ( - 2 π k m n ) p F ~ q ( a 1 , , a p TagBox[TagBox[TagBox[TagBox[RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; b 1 , , b q ; n q - p + 1 exp ( 2 π k n ) z 1 / n TagBox[RowBox[List[SuperscriptBox["n", RowBox[List["q", "-", "p", "+", "1"]]], " ", RowBox[List["exp", "(", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k"]], "n"], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "n"]]]]], HypergeometricPFQ, Rule[Editable, True]] ) /; m + n + η - q + 1 2 + m p + k = 1 q b k FormBox RowBox RowBox RowBox RowBox SubscriptBox ErrorBox FormBox RowBox RowBox n p + 1 TraditionalForm SubscriptBox OverscriptBox F ~ RowBox RowBox n q + n RowBox ( RowBox TagBox TagBox TagBox TagBox RowBox 1 , FractionBox RowBox SubscriptBox a 1 + m n , FractionBox RowBox SubscriptBox a 1 + m + 1 n , , FractionBox RowBox SubscriptBox a 1 + m + n - 1 n , , FractionBox RowBox SubscriptBox a p + m n , FractionBox RowBox SubscriptBox a p + m + 1 n , , FractionBox RowBox SubscriptBox a p + m + n - 1 n InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox TagBox TagBox RowBox FractionBox RowBox m + 1 n , FractionBox RowBox m + 2 n , , FractionBox RowBox m + n n , FractionBox RowBox SubscriptBox b 1 + m n , FractionBox RowBox SubscriptBox b 1 + m + 1 n , , FractionBox RowBox SubscriptBox b 1 + m + n - 1 n , , FractionBox RowBox SubscriptBox b q + m n , FractionBox RowBox SubscriptBox b q + m + 1 n , , FractionBox RowBox SubscriptBox b q + m + n - 1 n InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox z HypergeometricPFQ Rule Editable ) RowBox FractionBox RowBox SuperscriptBox RowBox ( RowBox 2 π ) RowBox FractionBox RowBox 1 - n 2 RowBox ( RowBox q + 1 ) SuperscriptBox n η RowBox UnderoverscriptBox RowBox j = 1 p TagBox SubscriptBox RowBox ( SubscriptBox a j ) m Pochhammer SuperscriptBox z RowBox - FractionBox m n RowBox UnderoverscriptBox RowBox k = 0 RowBox n - 1 RowBox RowBox exp ( RowBox - FractionBox RowBox 2 π k m n ) RowBox RowBox SubscriptBox FormBox p TraditionalForm SubscriptBox OverscriptBox F ~ q RowBox ( RowBox RowBox TagBox TagBox TagBox TagBox RowBox SubscriptBox a 1 , , SubscriptBox a p InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; SubscriptBox b 1 , , RowBox SubscriptBox b q ; TagBox RowBox SuperscriptBox n RowBox q - p + 1 RowBox exp ( FractionBox RowBox 2 π k n ) SuperscriptBox z RowBox 1 / n HypergeometricPFQ Rule Editable ) /; RowBox RowBox m SuperscriptBox + RowBox n SuperscriptBox + RowBox η RowBox RowBox - FractionBox RowBox q + 1 2 + RowBox m p + RowBox UnderoverscriptBox RowBox k = 1 q SubscriptBox b k TraditionalForm [/itex]

 Rule Form

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

 2001-10-29

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