html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.31.26.0005.01

 Input Form

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

 Standard Form

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 MathML Form

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

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

 2001-10-29