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 HypergeometricPFQ

 http://functions.wolfram.com/07.28.26.0005.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, 3}]/ (Pi Sin[Subscript[\[Psi], 3] Pi] Product[Gamma[Subscript[a, k]], {k, 1, 4}])) Sum[(Product[Sin[Pi (Subscript[b, j] - Subscript[a, k])], {k, 1, 4}]/Product[If[k != j, Sin[Pi (Subscript[b, j] - Subscript[b, k])], 1], {k, 1, 3}]) MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{0, 1 - Subscript[b, j]}, {1 - Subscript[b, 1], \[Ellipsis], 1 - Subscript[b, j - 1], 1 - Subscript[b, j + 1], \[Ellipsis], 1 - Subscript[b, 3]}}, z], {j, 1, 3}] - ((Pi Product[Gamma[Subscript[b, k]], {k, 1, 3}])/ (Sin[Subscript[\[Psi], 3] Pi] Product[Gamma[Subscript[a, k]], {k, 1, 4}])) (MeijerG[{{}, {1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2], 1 - Subscript[b, 3]}, {}}, z] + ((1 - z)^Subscript[\[Psi], 3] MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{}, {0, 1 - Subscript[b, 1], 1 - Subscript[b, 2], 1 - Subscript[b, 3]}}, z])/(z - 1)^Subscript[\[Psi], 3]) /; Subscript[\[Psi], 3] == Sum[Subscript[b, j], {j, 1, 3}] - Sum[Subscript[a, j], {j, 1, 4}] && !IntervalMemberQ[Interval[{-1, 0}], z] && !Element[Subscript[\[Psi], 3], Integers]

 Standard Form

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 MathML Form

 4 F 3 ( a 1 , a 2 , a 3 , a 4 ; b 1 , b 2 , b 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "3"], 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, False]], HypergeometricPFQ] k = 1 3 Γ ( b k ) π sin ( ψ 3 π ) k = 1 4 Γ ( a k ) j = 1 3 k = 1 4 sin ( π ( b j - a k ) ) k = 1 k j 3 sin ( π ( b j - b k ) ) G 4 , 4 2 , 4 ( z 1 - a 1 , 1 - a 2 , 1 - a 3 , 1 - a 4 0 , 1 - b j , 1 - b 1 , , 1 - b j - 1 , 1 - b j + 1 , , 1 - b q ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "4"]], RowBox[List["2", ",", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "3"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "4"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "j"]]], ",", RowBox[List["1", "-", SubscriptBox["b", "1"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["j", "-", "1"]]]]], MeijerG, Rule[Editable, True]]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["j", "+", "1"]]]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "q"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] - π k = 1 3 Γ ( b k ) sin ( ψ 3 π ) k = 1 4 Γ ( a k ) ( ( 1 - z ) ψ 3 ( z - 1 ) - ψ 3 G 4 , 4 0 , 4 ( z 1 - a 1 , 1 - a 2 , 1 - a 3 , 1 - a 4 0 , 1 - b 1 , 1 - b 2 , 1 - b 3 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "4"]], RowBox[List["0", ",", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "3"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "4"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "3"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] + G 4 , 4 4 , 0 ( z 1 - a 1 , 1 - a 2 , 1 - a 3 , 1 - a 4 0 , 1 - b 1 , 1 - b 2 , 1 - b 3 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "4"]], RowBox[List["4", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "3"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "4"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", "3"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] ) /; ψ 3 j = 1 3 b j - j = 1 4 a j z ( - 1 , 0 ) ψ 3 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] FormBox RowBox RowBox TagBox TagBox RowBox RowBox SubscriptBox FormBox 4 TraditionalForm SubscriptBox F FormBox 3 TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox SubscriptBox a 1 HypergeometricPFQ Rule Editable , TagBox SubscriptBox a 2 HypergeometricPFQ Rule Editable , TagBox SubscriptBox a 3 HypergeometricPFQ Rule Editable , TagBox SubscriptBox a 4 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox SubscriptBox b 1 HypergeometricPFQ Rule Editable , TagBox SubscriptBox b 2 HypergeometricPFQ Rule Editable , TagBox SubscriptBox b 3 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox z HypergeometricPFQ Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQ RowBox RowBox FractionBox RowBox UnderoverscriptBox RowBox k = 1 3 RowBox Γ RowBox ( SubscriptBox b k ) RowBox π RowBox sin RowBox ( RowBox SubscriptBox ψ 3 π ) RowBox UnderoverscriptBox RowBox k = 1 4 RowBox Γ RowBox ( SubscriptBox a k ) RowBox UnderoverscriptBox RowBox j = 1 3 ErrorBox RowBox FractionBox RowBox UnderoverscriptBox RowBox k = 1 4 RowBox sin RowBox ( RowBox π RowBox ( RowBox SubscriptBox b j - SubscriptBox a k ) ) RowBox UnderoverscriptBox UnderscriptBox RowBox k = 1 RowBox k j 3 RowBox sin RowBox ( RowBox π RowBox ( RowBox SubscriptBox b j - SubscriptBox b k ) ) TagBox RowBox SubsuperscriptBox TagBox G MeijerG RowBox 4 , 4 RowBox 2 , 4 RowBox ( RowBox TagBox z MeijerG Rule Editable GridBox RowBox TagBox RowBox 1 - SubscriptBox a 1 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a 2 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a 3 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a 4 MeijerG Rule Editable RowBox TagBox RowBox TagBox RowBox 0 , RowBox 1 - SubscriptBox b j , RowBox 1 - SubscriptBox b 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox j - 1 MeijerG Rule Editable MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox j + 1 MeijerG Rule Editable , TagBox MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b q MeijerG Rule Editable ) MeijerG Rule Editable - RowBox FractionBox RowBox π RowBox UnderoverscriptBox RowBox k = 1 3 RowBox Γ RowBox ( SubscriptBox b k ) RowBox RowBox sin RowBox ( RowBox SubscriptBox ψ 3 π ) RowBox UnderoverscriptBox RowBox k = 1 4 RowBox Γ RowBox ( SubscriptBox a k ) RowBox ( RowBox RowBox SuperscriptBox RowBox ( RowBox 1 - z ) SubscriptBox ψ 3 SuperscriptBox RowBox ( RowBox z - 1 ) RowBox - SubscriptBox ψ 3 TagBox RowBox SubsuperscriptBox TagBox G MeijerG RowBox 4 , 4 RowBox 0 , 4 RowBox ( RowBox TagBox z MeijerG Rule Editable GridBox RowBox TagBox RowBox 1 - SubscriptBox a 1 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a 2 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a 3 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a 4 MeijerG Rule Editable RowBox TagBox 0 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b 1 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b 2 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b 3 MeijerG Rule Editable ) MeijerG Rule Editable + TagBox RowBox SubsuperscriptBox TagBox G MeijerG RowBox 4 , 4 RowBox 4 , 0 RowBox ( RowBox TagBox z MeijerG Rule Editable GridBox RowBox TagBox RowBox 1 - SubscriptBox a 1 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a 2 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a 3 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a 4 MeijerG Rule Editable RowBox TagBox 0 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b 1 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b 2 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox b 3 MeijerG Rule Editable ) MeijerG Rule Editable ) /; RowBox RowBox SubscriptBox ψ 3 RowBox RowBox UnderoverscriptBox RowBox j = 1 3 SubscriptBox b j - RowBox UnderoverscriptBox RowBox j = 1 4 SubscriptBox a j RowBox z RowBox ( RowBox RowBox - 1 , 0 ) RowBox SubscriptBox ψ 3 TagBox Function TraditionalForm [/itex]

 Rule Form

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

 2001-10-29