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 HypergeometricPFQ

 http://functions.wolfram.com/07.26.06.0020.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2007-05-02