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/07.28.06.0005.01

 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}]/ Product[Gamma[Subscript[a, k]], {k, 1, 4}]) (Sum[Subscript[g, k][0] (1 - z)^k, {k, 0, Infinity}] + (1 - z)^Subscript[\[Psi], 3] Sum[Subscript[g, k][Subscript[\[Psi], 3]] (1 - z)^k, {k, 0, Infinity}]) /; Subscript[g, k][r] == ((-1)^k/k!) Gamma[Subscript[a, 1] + r + k] Gamma[Subscript[a, 2] + r + k] Gamma[Subscript[\[Psi], 3] - 2 r - k] Sum[((Pochhammer[Subscript[\[Psi], 3] - r - k, j] Pochhammer[Subscript[b, 2] + Subscript[b, 3] - Subscript[a, 3] - Subscript[a, 4], j] Pochhammer[Subscript[b, 1] - Subscript[a, 3], j])/(j! Gamma[Subscript[\[Psi], 3] + Subscript[a, 1] + j] Gamma[Subscript[\[Psi], 3] + Subscript[a, 2] + j])) HypergeometricPFQ[{Subscript[b, 3] - Subscript[a, 4], Subscript[b, 2] - Subscript[a, 4], -j}, {Subscript[b, 2] + Subscript[b, 3] - Subscript[a, 3] - Subscript[a, 4], 1 - Subscript[b, 1] + Subscript[a, 3] - j}, 1], {j, 0, Infinity}] && Subscript[\[Psi], 3] == Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3] - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] - Subscript[a, 4] && !Element[Subscript[\[Psi], 3], Integers] && Re[Subscript[a, 3]] > 0 && Re[Subscript[a, 4]] > 0

 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 ) k = 1 4 Γ ( a k ) ( ( 1 - z ) ψ 3 k = 0 g k ( ψ 3 ) ( 1 - z ) k + k = 0 g k ( 0 ) ( 1 - z ) k ) /; g k ( r ) ( - 1 ) k Γ ( k + r + a 1 ) Γ ( k + r + a 2 ) Γ ( ψ 3 - 2 r - k ) k ! j = 0 ( - k - r + ψ 3 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], "-", "r", "+", SubscriptBox["\[Psi]", "3"]]], ")"]], "j"], Pochhammer] ( - a 3 - a 4 + b 2 + b 3 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["a", "3"]]], "-", SubscriptBox["a", "4"], "+", SubscriptBox["b", "2"], "+", SubscriptBox["b", "3"]]], ")"]], "j"], Pochhammer] ( b 1 - a 3 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], ")"]], "j"], Pochhammer] j ! Γ ( j + a 1 + ψ 3 ) Γ ( j + a 2 + ψ 3 ) 3 F 2 ( b 3 - a 4 , b 2 - a 4 , - j ; b 2 + b 3 - a 3 - a 4 , a 3 - b 1 - j + 1 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "3"], "-", SubscriptBox["a", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "j"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "2"], "+", SubscriptBox["b", "3"], "-", SubscriptBox["a", "3"], "-", SubscriptBox["a", "4"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "-", "j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ψ 3 b 1 + b 2 + b 3 - a 1 - a 2 - a 3 - a 4 ψ 3 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Re ( a 3 ) > 0 Re ( a 4 ) > 0 Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript a 4 Subscript b 1 Subscript b 2 Subscript b 3 z k 1 3 Gamma Subscript b k k 1 4 Gamma Subscript a k -1 1 -1 z Subscript ψ 3 k 0 Subscript g k Subscript ψ 3 1 -1 z k k 0 Subscript g k 0 1 -1 z k Subscript g k r -1 k Gamma k r Subscript a 1 Gamma k r Subscript a 2 Gamma Subscript ψ 3 -1 2 r -1 k k -1 j 0 Pochhammer -1 k -1 r Subscript ψ 3 j Pochhammer -1 Subscript a 3 -1 Subscript a 4 Subscript b 2 Subscript b 3 j Pochhammer Subscript b 1 -1 Subscript a 3 j j Gamma j Subscript a 1 Subscript ψ 3 Gamma j Subscript a 2 Subscript ψ 3 -1 HypergeometricPFQ Subscript b 3 -1 Subscript a 4 Subscript b 2 -1 Subscript a 4 -1 j Subscript b 2 Subscript b 3 -1 Subscript a 3 -1 Subscript a 4 Subscript a 3 -1 Subscript b 1 -1 j 1 1 Subscript ψ 3 Subscript b 1 Subscript b 2 Subscript b 3 -1 Subscript a 1 -1 Subscript a 2 -1 Subscript a 3 -1 Subscript a 4 Subscript ψ 3 Subscript a 3 0 Subscript a 4 0 [/itex]

 Rule Form

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

 2001-10-29