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 HypergeometricPFQ

 http://functions.wolfram.com/07.29.06.0001.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4], Subscript[a, 5]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3], Subscript[b, 4]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, 4}]/Product[Gamma[Subscript[a, k]], {k, 1, 5}]) (Sum[Subscript[g, k][0] (1 - z)^k, {k, 0, Infinity}] + (1 - z)^Subscript[\[Psi], 4] Sum[Subscript[g, k][Subscript[\[Psi], 4]] (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], 4] - 2 r - k] Sum[((Pochhammer[Subscript[\[Psi], 4] - r - k, j] Pochhammer[Subscript[b, 2] + Subscript[b, 3] + Subscript[b, 4] - Subscript[a, 3] - Subscript[a, 4] - Subscript[a, 5], j] Pochhammer[Subscript[b, 1] - Subscript[a, 3], j])/ (Gamma[Subscript[\[Psi], 4] + Subscript[a, 1] + j] Gamma[Subscript[\[Psi], 4] + Subscript[a, 2] + j] j!)) Sum[((Pochhammer[Subscript[b, 4] + Subscript[b, 3] - Subscript[a, 5] - Subscript[a, 4], s] Pochhammer[Subscript[b, 2] - Subscript[a, 4], s] Pochhammer[-j, s])/(Pochhammer[Subscript[b, 2] + Subscript[b, 3] + Subscript[b, 4] - Subscript[a, 3] - Subscript[a, 4] - Subscript[a, 5], s] Pochhammer[ 1 - Subscript[b, 1] + Subscript[a, 3] - j, s] s!)) HypergeometricPFQ[{Subscript[b, 4] - Subscript[a, 5], Subscript[b, 3] - Subscript[a, 5], -s}, {Subscript[b, 4] + Subscript[b, 3] - Subscript[a, 5] - Subscript[a, 4], 1 - Subscript[b, 2] + Subscript[a, 4] - s}, 1], {s, 0, j}], {j, 0, Infinity}] && Subscript[\[Psi], 4] == Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3] + Subscript[b, 4] - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] - Subscript[a, 4] - Subscript[a, 5] && !Element[Subscript[\[Psi], 4], Integers]

 Standard Form

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 MathML Form

 5 F 4 ( a 1 , a 2 , a 3 , a 4 , a 5 ; b 1 , b 2 , b 3 , b 4 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["5", TraditionalForm]], SubscriptBox["F", FormBox["4", 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]], ",", TagBox[SubscriptBox["a", "5"], 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]], ",", TagBox[SubscriptBox["b", "4"], 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 4 Γ ( b k ) k = 1 5 Γ ( a k ) ( ( 1 - z ) ψ 4 k = 0 g k ( ψ 4 ) ( 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 4 - 2 r - k ) k ! j = 0 ( ψ 4 - r - k ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["\[Psi]", "4"], "-", "r", "-", "k"]], ")"]], "j"], Pochhammer] ( b 2 + b 3 + b 4 - a 3 - a 4 - a 5 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "+", SubscriptBox["b", "3"], "+", SubscriptBox["b", "4"], "-", SubscriptBox["a", "3"], "-", SubscriptBox["a", "4"], "-", SubscriptBox["a", "5"]]], ")"]], "j"], Pochhammer] ( b 1 - a 3 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], ")"]], "j"], Pochhammer] Γ ( j + a 1 + ψ 4 ) Γ ( j + a 2 + ψ 4 ) j ! s = 0 j ( b 3 + b 4 - a 4 - a 5 ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "3"], "+", SubscriptBox["b", "4"], "-", SubscriptBox["a", "4"], "-", SubscriptBox["a", "5"]]], ")"]], "s"], Pochhammer] ( b 2 - a 4 ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "4"]]], ")"]], "s"], Pochhammer] ( - j ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "j"]], ")"]], "s"], Pochhammer] ( b 2 + b 3 + b 4 - a 3 - a 4 - a 5 ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "+", SubscriptBox["b", "3"], "+", SubscriptBox["b", "4"], "-", SubscriptBox["a", "3"], "-", SubscriptBox["a", "4"], "-", SubscriptBox["a", "5"]]], ")"]], "s"], Pochhammer] ( a 3 - b 1 - j + 1 ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "-", "j", "+", "1"]], ")"]], "s"], Pochhammer] s ! 3 F 2 ( b 4 - a 5 , b 3 - a 5 , - s ; b 3 + b 4 - a 4 - a 5 , a 4 - b 2 - s + 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", "4"], "-", SubscriptBox["a", "5"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "3"], "-", SubscriptBox["a", "5"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "s"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "3"], "+", SubscriptBox["b", "4"], "-", SubscriptBox["a", "4"], "-", SubscriptBox["a", "5"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "4"], "-", SubscriptBox["b", "2"], "-", "s", "+", "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 4 b 1 + b 2 + b 3 + b 4 - a 1 - a 2 - a 3 - a 4 - a 5 ψ 4 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript a 4 Subscript a 5 Subscript b 1 Subscript b 2 Subscript b 3 Subscript b 4 z k 1 4 Gamma Subscript b k k 1 5 Gamma Subscript a k -1 1 -1 z Subscript ψ 4 k 0 Subscript g k Subscript ψ 4 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 4 -1 2 r -1 k k -1 j 0 Pochhammer Subscript ψ 4 -1 r -1 k j Pochhammer Subscript b 2 Subscript b 3 Subscript b 4 -1 Subscript a 3 -1 Subscript a 4 -1 Subscript a 5 j Pochhammer Subscript b 1 -1 Subscript a 3 j Gamma j Subscript a 1 Subscript ψ 4 Gamma j Subscript a 2 Subscript ψ 4 j -1 s 0 j Pochhammer Subscript b 3 Subscript b 4 -1 Subscript a 4 -1 Subscript a 5 s Pochhammer Subscript b 2 -1 Subscript a 4 s Pochhammer -1 j s Pochhammer Subscript b 2 Subscript b 3 Subscript b 4 -1 Subscript a 3 -1 Subscript a 4 -1 Subscript a 5 s Pochhammer Subscript a 3 -1 Subscript b 1 -1 j 1 s s -1 HypergeometricPFQ Subscript b 4 -1 Subscript a 5 Subscript b 3 -1 Subscript a 5 -1 s Subscript b 3 Subscript b 4 -1 Subscript a 4 -1 Subscript a 5 Subscript a 4 -1 Subscript b 2 -1 s 1 1 Subscript ψ 4 Subscript b 1 Subscript b 2 Subscript b 3 Subscript b 4 -1 Subscript a 1 -1 Subscript a 2 -1 Subscript a 3 -1 Subscript a 4 -1 Subscript a 5 Subscript ψ 4 [/itex]

 Rule Form

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

 2001-10-29