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 HypergeometricPFQ

 http://functions.wolfram.com/07.28.06.0009.01

 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}]) ((1 - z)^Subscript[\[Psi], 3] Sum[(-Subscript[\[Psi], 3] - j - 1)! Pochhammer[Subscript[\[Psi], 3] + Subscript[a, 1], j] Pochhammer[Subscript[\[Psi], 3] + Subscript[a, 2], j] Sum[(((-1)^k HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, k])/ ((j - k)! Pochhammer[Subscript[\[Psi], 3] + Subscript[a, 1], k] Pochhammer[Subscript[\[Psi], 3] + Subscript[a, 2], k])) (z - 1)^j, {k, 0, j}], {j, 0, -Subscript[\[Psi], 3] - 1}] + Sum[(1/(j - Subscript[\[Psi], 3])!) Sum[((Gamma[Subscript[a, 1] + k + j] Gamma[Subscript[a, 2] + k + j] HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, k])/ ((k + j)! Gamma[Subscript[\[Psi], 3] + Subscript[a, 1] + k] Gamma[Subscript[\[Psi], 3] + Subscript[a, 2] + k])) (-Log[1 - z] + PolyGamma[k + j + 1] + PolyGamma[ j - Subscript[\[Psi], 3] + 1] - PolyGamma[Subscript[a, 1] + k + j] - PolyGamma[Subscript[a, 2] + k + j]) (1 - z)^j, {k, 0, Infinity}], {j, Subscript[\[Psi], 3], Infinity}]/ (-1)^Subscript[\[Psi], 3] + (-1)^(-Subscript[\[Psi], 3] - 1) Sum[((Gamma[Subscript[a, 1] + k + j] Gamma[Subscript[a, 2] + k + j] HypergeometricPFQExpansionCoefficient[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, k])/ ((k + j)! Gamma[Subscript[\[Psi], 3] + Subscript[a, 1] + k] Gamma[Subscript[\[Psi], 3] + Subscript[a, 2] + k])) (1 - z)^j, {k, 0, Infinity}, {j, -k, Subscript[\[Psi], 3] - 1}]) /; Abs[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] && -Subscript[\[Psi], 3] > 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 j = 0 - ψ 3 - 1 ( - j - ψ 3 - 1 ) ! ( a 1 + ψ 3 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]], "j"], Pochhammer] ( a 2 + ψ 3 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]], "j"], Pochhammer] k = 0 j ( - 1 ) k k ( 3 ) ( { a 1 , a 2 , a 3 , a 4 } , { b 1 , b 2 , b 3 } ) ( z - 1 ) j ( j - k ) ! ( a 1 + ψ 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]], "k"], Pochhammer] ( a 2 + ψ 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "3"]]], ")"]], "k"], Pochhammer] + ( - 1 ) - ψ 3 j = ψ 3 1 ( j - ψ 3 ) ! k = 0 Γ ( j + k + a 1 ) Γ ( j + k + a 2 ) ( j + k ) ! Γ ( k + a 1 + ψ 3 ) Γ ( k + a 2 + ψ 3 ) k ( 3 ) ( { a 1 , a 2 , a 3 , a 4 } , { b 1 , b 2 , b 3 } ) ( - log ( 1 - z ) + ψ TagBox["\[Psi]", PolyGamma] ( j + k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( j - ψ 3 + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + k + a 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + k + a 2 ) ) ( 1 - z ) j + ( - 1 ) - ψ 3 - 1 k = 0 j = - k ψ 3 - 1 Γ ( j + k + a 1 ) Γ ( j + k + a 2 ) ( j + k ) ! Γ ( k + a 1 + ψ 3 ) Γ ( k + a 2 + ψ 3 ) k ( 3 ) ( { a 1 , a 2 , a 3 , a 4 } , { b 1 , b 2 , b 3 } ) ( 1 - z ) j ) /; "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 1 ψ 3 j = 1 3 b j - j = 1 4 a j - ψ 3 + 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 j 0 -1 Subscript ψ 3 -1 -1 j -1 Subscript ψ 3 -1 Pochhammer Subscript a 1 Subscript ψ 3 j Pochhammer Subscript a 2 Subscript ψ 3 j k 0 j -1 k Subscript k 3 Subscript a 1 Subscript a 2 Subscript a 3 Subscript a 4 Subscript b 1 Subscript b 2 Subscript b 3 z -1 j j -1 k Pochhammer Subscript a 1 Subscript ψ 3 k Pochhammer Subscript a 2 Subscript ψ 3 k -1 -1 -1 Subscript ψ 3 j Subscript ψ 3 1 j -1 Subscript ψ 3 -1 k 0 Gamma j k Subscript a 1 Gamma j k Subscript a 2 j k Gamma k Subscript a 1 Subscript ψ 3 Gamma k Subscript a 2 Subscript ψ 3 -1 Subscript k 3 Subscript a 1 Subscript a 2 Subscript a 3 Subscript a 4 Subscript b 1 Subscript b 2 Subscript b 3 -1 1 -1 z PolyGamma j k 1 PolyGamma j -1 Subscript ψ 3 1 -1 PolyGamma j k Subscript a 1 -1 PolyGamma j k Subscript a 2 1 -1 z j -1 -1 Subscript ψ 3 -1 j -1 k Subscript ψ 3 -1 k 0 Gamma j k Subscript a 1 Gamma j k Subscript a 2 j k Gamma k Subscript a 1 Subscript ψ 3 Gamma k Subscript a 2 Subscript ψ 3 -1 Subscript k 3 Subscript a 1 Subscript a 2 Subscript a 3 Subscript a 4 Subscript b 1 Subscript b 2 Subscript b 3 1 -1 z j 1 -1 z 1 Subscript ψ 3 j 1 3 Subscript b j -1 j 1 4 Subscript a j -1 Subscript ψ 3 SuperPlus [/itex]

 Rule Form

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

 2001-10-29