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 HypergeometricPFQ

 http://functions.wolfram.com/07.28.06.0012.01

 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] \[Proportional] (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]}}, 1] (1 + O[z - 1]), {j, 1, 3}] + Gamma[-Subscript[\[Psi], 3]] (Product[Gamma[Subscript[b, k]], {k, 1, 3}]/ Product[Gamma[Subscript[a, k]], {k, 1, 4}]) (1 - z)^Subscript[\[Psi], 3] (1 + O[z - 1]) /; (z -> 1) && Subscript[\[Psi], 3] == Sum[Subscript[b, j], {j, 1, 3}] - Sum[Subscript[a, j], {j, 1, 4}] && !Element[Subscript[\[Psi], 3], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["a", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]]]], RowBox[List["\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], " ", "\[Pi]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "4"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "3"], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "4"], RowBox[List["Sin", "[", RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["a", "k"]]], ")"]]]], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List["If", "[", RowBox[List[RowBox[List["k", "\[NotEqual]", "j"]], ",", RowBox[List["Sin", "[", RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], ")"]]]], "]"]], ",", "1"]], "]"]]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "-", SubscriptBox["a", "2"]]], ",", RowBox[List["1", "-", SubscriptBox["a", "3"]]], ",", RowBox[List["1", "-", SubscriptBox["a", "4"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List["1", "-", SubscriptBox["b", "j"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", RowBox[List["j", "-", "1"]]]]], ",", RowBox[List["1", "-", SubscriptBox["b", RowBox[List["j", "+", "1"]]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "3"]]]]], "}"]]]], "}"]], ",", "1"]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]]]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", SubscriptBox["\[Psi]", "3"]]], "]"]], FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "3"], RowBox[List["Gamma", "[", SubscriptBox["b", "k"], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "4"], RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], SubscriptBox["\[Psi]", "3"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", RowBox[List["z", "-", "1"]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "1"]], ")"]], "\[And]", RowBox[List[SubscriptBox["\[Psi]", "3"], "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "3"], SubscriptBox["b", "j"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "4"], SubscriptBox["a", "j"]]]]]]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[SubscriptBox["\[Psi]", "3"], ",", "Integers"]], "]"]], "]"]]]]]]]]

 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] Γ ( - ψ 3 ) k = 1 3 Γ ( b k ) k = 1 4 Γ ( a k ) ( 1 - z ) ψ 3 ( 1 + O ( z - 1 ) ) + 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 ( 1 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 3 ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "4"]], RowBox[List["2", ",", "4"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["1", 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]], ",", RowBox[List["1", "-", SubscriptBox["a", "3"]]], ",", 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", "3"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] ( 1 + O ( z - 1 ) ) /; ( z "\[Rule]" 1 ) ψ 3 j = 1 3 b j - j = 1 4 a j 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 RowBox Γ ( RowBox - SubscriptBox ψ 3 ) RowBox UnderoverscriptBox RowBox k = 1 3 RowBox Γ ( SubscriptBox b k ) RowBox UnderoverscriptBox RowBox k = 1 4 RowBox Γ ( SubscriptBox a k ) SuperscriptBox RowBox ( RowBox 1 - z ) SubscriptBox ψ 3 RowBox ( RowBox 1 + RowBox O ( RowBox z - 1 ) ) + RowBox FractionBox RowBox UnderoverscriptBox RowBox k = 1 3 RowBox Γ ( SubscriptBox b k ) RowBox π RowBox sin ( RowBox SubscriptBox ψ 3 π ) RowBox UnderoverscriptBox RowBox k = 1 4 RowBox Γ ( SubscriptBox a k ) RowBox UnderoverscriptBox RowBox j = 1 3 ErrorBox RowBox FractionBox RowBox UnderoverscriptBox RowBox k = 1 4 RowBox sin ( RowBox π RowBox ( RowBox SubscriptBox b j - SubscriptBox a k ) ) RowBox UnderoverscriptBox UnderscriptBox RowBox k = 1 RowBox k j 3 RowBox sin ( 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 1 MeijerG Rule Editable GridBox RowBox TagBox RowBox 1 - SubscriptBox a 1 MeijerG Rule Editable , TagBox RowBox 1 - SubscriptBox a 2 MeijerG Rule Editable , RowBox 1 - SubscriptBox a 3 , 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 3 MeijerG Rule Editable ) MeijerG Rule Editable RowBox ( RowBox 1 + RowBox O ( RowBox z - 1 ) ) /; RowBox RowBox ( RowBox z 1 ) RowBox SubscriptBox ψ 3 RowBox RowBox UnderoverscriptBox RowBox j = 1 3 SubscriptBox b j - RowBox UnderoverscriptBox RowBox j = 1 4 SubscriptBox a j 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