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 HypergeometricPFQ

 http://functions.wolfram.com/07.28.06.0007.01

 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}] + 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] Sum[(((-1)^k Subscript[c, k])/Pochhammer[Subscript[\[Psi], 3] + 1, k]) (z - 1)^k, {k, 0, Infinity}] /; Abs[z - 1] < 1 && Subscript[\[Psi], 3] == Sum[Subscript[b, j], {j, 1, 3}] - Sum[Subscript[a, j], {j, 1, 4}] && Q[t] == Product[t - 1 + Subscript[b, k], {k, 1, 3}] && R[t] == Product[t + Subscript[a, k], {k, 1, 4}] && Subscript[\[CapitalDelta], n][f[x]] == Sum[(-1)^(n - k) Binomial[n, k] f[x + k], {k, 0, n}] && (Subscript[c, k] == 0 /; k < 0) && Subscript[c, 0] == 1 && Subscript[c, 1] == (-(\[CapitalDelta][Q[Subscript[\[Psi], 3]]] - Subscript[\[CapitalDelta], 2][R[Subscript[\[Psi], 3] - 1]]/2)) Subscript[c, 0] && Subscript[c, k] == (1/k) (R[Subscript[\[Psi], 3] + k - 3] Subscript[c, k - 3] + Sum[(Subscript[\[CapitalDelta], j - 1][Q[Subscript[\[Psi], 3] + k - 2]]/ (j - 1)! - Subscript[\[CapitalDelta], j][ R[Subscript[\[Psi], 3] + k - 3]]/j!) Subscript[c, k + j - 3], {j, 1, 2}]) && !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] Γ ( - ψ 3 ) k = 1 3 Γ ( b k ) k = 1 4 Γ ( a k ) ( 1 - z ) ψ 3 k = 0 ( - 1 ) k c k ( z - 1 ) k ( ψ 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["\[Psi]", "3"], "+", "1"]], ")"]], "k"], Pochhammer] + 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 3 ) 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]], ",", 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]] /; "\[LeftBracketingBar]" z - 1 "\[RightBracketingBar]" < 1 ψ 3 j = 1 3 b j - j = 1 4 a j Q ( t ) k = 1 3 ( t + b k - 1 ) R ( t ) k = 1 4 ( t + a k ) Δ n ( f ( x ) ) k = 0 n ( - 1 ) n - k ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] f ( k + x ) ( c k 0 /; k < 0 ) c 0 1 c 1 - ( Δ ( Q ( ψ 3 ) ) - 1 2 Δ 2 ( R ( ψ 3 - 1 ) ) ) c 0 c k 1 k ( R ( k + ψ 3 - 3 ) c k - 3 + j = 1 2 ( Δ j - 1 ( Q ( k + ψ 3 - 2 ) ) ( j - 1 ) ! - Δ j ( R ( k + ψ 3 - 3 ) ) j ! ) c j + k - 3 ) 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 UnderoverscriptBox RowBox k = 0 FractionBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) k SubscriptBox c k SuperscriptBox RowBox ( RowBox z - 1 ) k TagBox SubscriptBox RowBox ( RowBox SubscriptBox ψ 3 + 1 ) k Pochhammer + 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 z 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 RowBox RowBox z - 1 < 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 RowBox Q ( t ) RowBox UnderoverscriptBox RowBox k = 1 3 RowBox ( RowBox t + SubscriptBox b k - 1 ) RowBox RowBox R ( t ) RowBox UnderoverscriptBox RowBox k = 1 4 RowBox ( RowBox t + SubscriptBox a k ) RowBox RowBox SubscriptBox Δ n ( RowBox f ( x ) ) RowBox UnderoverscriptBox RowBox k = 0 n RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox n - k TagBox RowBox ( GridBox TagBox n Rule Editable TagBox k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox f ( RowBox k + x ) RowBox ( RowBox RowBox SubscriptBox c k 0 /; RowBox k < 0 ) RowBox SubscriptBox c 0 1 RowBox SubscriptBox c 1 RowBox RowBox - RowBox ( RowBox RowBox Δ ( RowBox Q ( SubscriptBox ψ 3 ) ) - RowBox FractionBox 1 2 RowBox SubscriptBox Δ 2 ( RowBox R ( RowBox SubscriptBox ψ 3 - 1 ) ) ) SubscriptBox c 0 RowBox SubscriptBox c k RowBox FractionBox 1 k RowBox ( RowBox RowBox RowBox R ( RowBox k + SubscriptBox ψ 3 - 3 ) SubscriptBox c RowBox k - 3 + RowBox UnderoverscriptBox RowBox j = 1 2 RowBox RowBox ( RowBox FractionBox RowBox SubscriptBox Δ RowBox j - 1 ( RowBox Q ( RowBox k + SubscriptBox ψ 3 - 2 ) ) RowBox RowBox ( RowBox j - 1 ) ! - FractionBox RowBox SubscriptBox Δ j ( RowBox R ( RowBox k + SubscriptBox ψ 3 - 3 ) ) RowBox j ! ) SubscriptBox c RowBox j + k - 3 ) 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