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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.21.0003.01

 Input Form

 Integrate[t^(\[Alpha] - 1) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, -t], {t, 0, Infinity}] == (Product[Gamma[Subscript[b, k]], {k, 1, q}]/ Product[Gamma[Subscript[a, k]], {k, 1, p}]) ((Gamma[\[Alpha]] Product[Gamma[Subscript[a, k] - \[Alpha]], {k, 1, p}])/ Product[Gamma[Subscript[b, k] - \[Alpha]], {k, 1, q}]) /; (0 < Re[\[Alpha]] < Min[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, p]]] && p - 1 <= q <= p) || (0 < Re[\[Alpha]] < Min[Re[Subscript[a, 1]], \[Ellipsis], Re[Subscript[a, p]], 1/4 - (1/2) Re[Sum[Subscript[a, j], {j, 1, p}] - Sum[Subscript[b, k], {k, 1, q}]]] && q == p + 1)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", RowBox[List["-", "t"]]]], "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]], RowBox[List[" ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]]]], FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", "\[Alpha]", "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "-", "\[Alpha]"]], "]"]]]]]]]], RowBox[List[" ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "q"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "k"], "-", "\[Alpha]"]], "]"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["0", "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["a", "p"], "]"]]]], "]"]]]], "\[And]", RowBox[List[RowBox[List["p", "-", "1"]], "\[LessEqual]", "q", "\[LessEqual]", "p"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["0", "<", RowBox[List["Re", "[", "\[Alpha]", "]"]], "<", RowBox[List["Min", "[", RowBox[List[RowBox[List["Re", "[", SubscriptBox["a", "1"], "]"]], ",", "\[Ellipsis]", ",", RowBox[List["Re", "[", SubscriptBox["a", "p"], "]"]], ",", RowBox[List[FractionBox["1", "4"], "-", RowBox[List[FractionBox["1", "2"], RowBox[List["Re", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["a", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], SubscriptBox["b", "k"]]]]], "]"]]]]]]]], "]"]]]], "\[And]", RowBox[List["q", "\[Equal]", RowBox[List["p", "+", "1"]]]]]], ")"]]]]]]]]

 MathML Form

 0 t α - 1 p F q ( a 1 , , a p TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; b 1 TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]] , TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]] , b q TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]] ; - t TagBox[RowBox[List["-", "t"]], HypergeometricPFQ, Rule[Editable, True]] ) t ( k = 1 q Γ ( b k ) ) ( Γ ( α ) k = 1 p Γ ( a k - α ) ) ( k = 1 p Γ ( a k ) ) k = 1 q Γ ( b k - α ) /; 0 < Re ( α ) < min ( Re ( a 1 ) , , Re ( a p ) ) p - 1 q p 0 < Re ( α ) < min ( Re ( a 1 ) , , Re ( a p ) , 1 4 - 1 2 Re ( j = 1 p a j - k = 1 q b k ) ) q p + 1 FormBox RowBox RowBox RowBox SubsuperscriptBox 0 RowBox SuperscriptBox t RowBox α - 1 RowBox RowBox SubscriptBox ErrorBox p SubscriptBox F FormBox q TraditionalForm RowBox ( RowBox RowBox TagBox TagBox RowBox TagBox SubscriptBox a 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox SubscriptBox a p HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox SubscriptBox b 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , RowBox TagBox SubscriptBox b q HypergeometricPFQ Rule Editable ; TagBox RowBox - t HypergeometricPFQ Rule Editable ) RowBox t FractionBox RowBox RowBox ( RowBox UnderoverscriptBox RowBox k = 1 q RowBox Γ ( SubscriptBox b k ) ) RowBox ( RowBox RowBox Γ ( α ) RowBox UnderoverscriptBox RowBox k = 1 p RowBox Γ ( RowBox SubscriptBox a k - α ) ) RowBox RowBox ( RowBox UnderoverscriptBox RowBox k = 1 p RowBox Γ ( SubscriptBox a k ) ) RowBox UnderoverscriptBox RowBox k = 1 q RowBox Γ ( RowBox SubscriptBox b k - α ) /; RowBox RowBox RowBox 0 < RowBox Re ( α ) < RowBox min ( RowBox RowBox Re ( SubscriptBox a 1 ) , , RowBox Re ( SubscriptBox a p ) ) RowBox RowBox p - 1 q p RowBox RowBox 0 < RowBox Re ( α ) < RowBox min ( RowBox RowBox Re ( SubscriptBox a 1 ) , , RowBox Re ( SubscriptBox a p ) , RowBox FractionBox 1 4 - RowBox FractionBox 1 2 RowBox Re ( RowBox RowBox UnderoverscriptBox RowBox j = 1 p SubscriptBox a j - RowBox UnderoverscriptBox RowBox k = 1 q SubscriptBox b k ) ) RowBox q RowBox p + 1 TraditionalForm [/itex]

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

 2001-10-29