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 HypergeometricPFQ

 http://functions.wolfram.com/07.27.06.0040.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] == Subscript[F, Infinity][z, Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[b, 1], Subscript[b, 2]] /; (Subscript[F, n][z, Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[b, 1], Subscript[b, 2]] == ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]])) ((((Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2] - Subscript[a, 1]] Gamma[Subscript[a, 3] - Subscript[a, 1]])/ (Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]])) Sum[(Pochhammer[Subscript[a, 1], k] Pochhammer[1 + Subscript[a, 1] - Subscript[b, 1], k] Pochhammer[1 + Subscript[a, 1] - Subscript[b, 2], k])/z^k/(k! Pochhammer[1 + Subscript[a, 1] - Subscript[a, 2], k] Pochhammer[1 + Subscript[a, 1] - Subscript[a, 3], k]), {k, 0, n}])/(-z)^Subscript[a, 1] + (((Gamma[Subscript[a, 2]] Gamma[Subscript[a, 1] - Subscript[a, 2]] Gamma[Subscript[a, 3] - Subscript[a, 2]])/ (Gamma[Subscript[b, 1] - Subscript[a, 2]] Gamma[Subscript[b, 2] - Subscript[a, 2]])) Sum[(Pochhammer[Subscript[a, 2], k] Pochhammer[1 + Subscript[a, 2] - Subscript[b, 1], k] Pochhammer[1 + Subscript[a, 2] - Subscript[b, 2], k])/z^k/(k! Pochhammer[1 + Subscript[a, 2] - Subscript[a, 1], k] Pochhammer[1 + Subscript[a, 2] - Subscript[a, 3], k]), {k, 0, n}])/(-z)^Subscript[a, 2] + (((Gamma[Subscript[a, 3]] Gamma[Subscript[a, 1] - Subscript[a, 3]] Gamma[Subscript[a, 2] - Subscript[a, 3]])/ (Gamma[Subscript[b, 1] - Subscript[a, 3]] Gamma[Subscript[b, 2] - Subscript[a, 3]])) Sum[(Pochhammer[Subscript[a, 3], k] Pochhammer[1 + Subscript[a, 3] - Subscript[b, 1], k] Pochhammer[1 + Subscript[a, 3] - Subscript[b, 2], k])/z^k/(k! Pochhammer[1 + Subscript[a, 3] - Subscript[a, 1], k] Pochhammer[1 + Subscript[a, 3] - Subscript[a, 2], k]), {k, 0, n}])/(-z)^Subscript[a, 3]) == HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] - ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]])) (((Sin[(Subscript[b, 1] - Subscript[a, 1]) Pi] Sin[(Subscript[b, 2] - Subscript[a, 1]) Pi] Gamma[1 + Subscript[a, 1] + n] Gamma[2 + Subscript[a, 1] - Subscript[b, 1] + n] Gamma[2 + Subscript[a, 1] - Subscript[b, 2] + n] z^(-n - 1))/(-z)^Subscript[a, 1]/ (Sin[(Subscript[a, 2] - Subscript[a, 1]) Pi] Sin[(Subscript[a, 3] - Subscript[a, 1]) Pi] Gamma[2 + Subscript[a, 1] - Subscript[a, 2] + n] Gamma[2 + Subscript[a, 1] - Subscript[a, 3] + n] (n + 1)!)) HypergeometricPFQ[{1, 1 + Subscript[a, 1] + n, 2 + Subscript[a, 1] - Subscript[b, 1] + n, 2 + Subscript[a, 1] - Subscript[b, 2] + n}, {2 + n, 2 + Subscript[a, 1] - Subscript[a, 2] + n, 2 + Subscript[a, 1] - Subscript[a, 3] + n}, 1/z] + ((Sin[(Subscript[b, 1] - Subscript[a, 2]) Pi] Sin[(Subscript[b, 2] - Subscript[a, 2]) Pi] Gamma[1 + Subscript[a, 2] + n] Gamma[2 + Subscript[a, 2] - Subscript[b, 1] + n] Gamma[2 + Subscript[a, 2] - Subscript[b, 2] + n] z^(-n - 1))/(-z)^Subscript[a, 2]/ (Sin[(Subscript[a, 1] - Subscript[a, 2]) Pi] Sin[(Subscript[a, 3] - Subscript[a, 2]) Pi] Gamma[2 + Subscript[a, 2] - Subscript[a, 1] + n] Gamma[2 + Subscript[a, 2] - Subscript[a, 3] + n] (n + 1)!)) HypergeometricPFQ[{1, 1 + Subscript[a, 2] + n, 2 + Subscript[a, 2] - Subscript[b, 1] + n, 2 + Subscript[a, 2] - Subscript[b, 2] + n}, {2 + n, 2 + Subscript[a, 2] - Subscript[a, 1] + n, 2 + Subscript[a, 2] - Subscript[a, 3] + n}, 1/z] + ((Sin[(Subscript[b, 1] - Subscript[a, 3]) Pi] Sin[(Subscript[b, 2] - Subscript[a, 3]) Pi] Gamma[1 + Subscript[a, 3] + n] Gamma[2 + Subscript[a, 3] - Subscript[b, 1] + n] Gamma[2 + Subscript[a, 3] - Subscript[b, 2] + n] z^(-n - 1))/(-z)^Subscript[a, 3]/ (Sin[(Subscript[a, 2] - Subscript[a, 3]) Pi] Sin[(Subscript[a, 1] - Subscript[a, 3]) Pi] Gamma[2 + Subscript[a, 3] - Subscript[a, 2] + n] Gamma[2 + Subscript[a, 3] - Subscript[a, 1] + n] (n + 1)!)) HypergeometricPFQ[{1, 1 + Subscript[a, 3] + n, 2 + Subscript[a, 3] - Subscript[b, 1] + n, 2 + Subscript[a, 3] - Subscript[b, 2] + n}, {2 + n, 2 + Subscript[a, 3] - Subscript[a, 2] + n, 2 + Subscript[a, 3] - Subscript[a, 1] + n}, 1/z]) && Element[n, Integers] && n >= 0) && !Element[Subscript[a, 1] - Subscript[a, 2], Integers] && !Element[Subscript[a, 1] - Subscript[a, 3], Integers] && !Element[Subscript[a, 2] - Subscript[a, 3], Integers]

 Standard Form

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 MathML Form

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( a 1 - a 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 1 - a 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"], "+", "1"]], ")"]], "k"], Pochhammer] + Γ ( a 2 ) Γ ( a 1 - a 2 ) Γ ( a 3 - a 2 ) ( - z ) - a 2 Γ ( b 1 - a 2 ) Γ ( b 2 - a 2 ) k = 0 n ( a 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "2"], ")"]], "k"], Pochhammer] ( a 2 - b 1 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 2 - b 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"], "+", "1"]], ")"]], "k"], Pochhammer] z - k k ! ( - a 1 + a 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["a", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 2 - a 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"], "+", "1"]], ")"]], "k"], Pochhammer] + Γ ( a 3 ) Γ ( a 1 - a 3 ) Γ ( a 2 - a 3 ) ( - z ) - a 3 Γ ( b 1 - a 3 ) Γ ( b 2 - a 3 ) k = 0 n ( a 3 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "3"], ")"]], "k"], Pochhammer] ( a 3 - b 1 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 3 - b 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"], "+", "1"]], ")"]], "k"], Pochhammer] z - k k ! 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( a 1 - a 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 1 - a 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"], "+", "1"]], ")"]], "k"], Pochhammer] + Γ ( a 2 ) Γ ( a 1 - a 2 ) Γ ( a 3 - a 2 ) ( - z ) - a 2 Γ ( b 1 - a 2 ) Γ ( b 2 - a 2 ) k = 0 n ( a 2 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "2"], ")"]], "k"], Pochhammer] ( a 2 - b 1 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 2 - b 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"], "+", "1"]], ")"]], "k"], Pochhammer] z - k k ! ( - a 1 + a 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["a", "2"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 2 - a 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"], "+", "1"]], ")"]], "k"], Pochhammer] + Γ ( a 3 ) Γ ( a 1 - a 3 ) Γ ( a 2 - a 3 ) ( - z ) - a 3 Γ ( b 1 - a 3 ) Γ ( b 2 - a 3 ) k = 0 n ( a 3 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "3"], ")"]], "k"], Pochhammer] ( a 3 - b 1 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "+", "1"]], ")"]], "k"], Pochhammer] ( a 3 - b 2 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"], "+", "1"]], ")"]], "k"], Pochhammer] z - k k ! ( - a 1 + a 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["a", "1"]]], "+", SubscriptBox["a", "3"], "+", "1"]], ")"]], "k"], Pochhammer] ( - a 2 + a 3 + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SubscriptBox["a", "2"]]], "+", SubscriptBox["a", "3"], "+", "1"]], ")"]], "k"], Pochhammer] ) 3 F 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; z ) 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[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], 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]]]], 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 ( b 1 ) Γ ( b 2 ) Γ ( a 1 ) Γ ( a 2 ) Γ ( a 3 ) ( ( sin ( ( b 1 - a 1 ) π ) sin ( ( b 2 - a 1 ) π ) Γ ( n + a 1 + 1 ) Γ ( n + a 1 - b 1 + 2 ) Γ ( n + a 1 - b 2 + 2 ) ( - z ) - a 1 z - n - 1 ) / ( sin ( ( a 2 - a 1 ) π ) sin ( ( a 3 - a 1 ) π ) Γ ( n + a 1 - a 2 + 2 ) Γ ( n + a 1 - a 3 + 2 ) ( n + 1 ) ! 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02