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 HypergeometricPFQ

 http://functions.wolfram.com/07.26.06.0010.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] \[Proportional] ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]])) (((-z)^\[Chi]/Sqrt[Pi]) (Cos[2 Sqrt[-z] + Pi \[Chi]] (1 + O[1/z]) + (1/(16 Sqrt[-z])) (16 (Subscript[b, 1] Subscript[b, 2] + Subscript[b, 1] Subscript[b, 3] + Subscript[b, 2] Subscript[b, 3] - Subscript[a, 1] Subscript[a, 2]) + 2 (4 \[Chi] - 1) (3 Subscript[a, 1] + 3 Subscript[a, 2] + Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3] - 2) - 3) Sin[2 Sqrt[-z] + Pi \[Chi]] (1 + O[1/z])) + (((Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2] - Subscript[a, 1]])/ (Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]] Gamma[Subscript[b, 3] - Subscript[a, 1]])) (1 + O[1/z]))/ (-z)^Subscript[a, 1] + (((Gamma[Subscript[a, 2]] Gamma[Subscript[a, 1] - Subscript[a, 2]])/ (Gamma[Subscript[b, 1] - Subscript[a, 2]] Gamma[Subscript[b, 2] - Subscript[a, 2]] Gamma[Subscript[b, 3] - Subscript[a, 2]])) (1 + O[1/z]))/ (-z)^Subscript[a, 2]) /; \[Chi] == (1/2) (Subscript[a, 1] + Subscript[a, 2] - Subscript[b, 1] - Subscript[b, 2] - Subscript[b, 3] + 1/2) && Subscript[a, 1] != Subscript[a, 2] && (Abs[z] -> Infinity)

 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[RowBox[List["-", "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 ( b 3 ) Γ ( a 1 ) Γ ( a 2 ) ( Γ ( a 1 ) Γ ( a 2 - a 1 ) Γ ( b 1 - a 1 ) Γ ( b 2 - a 1 ) Γ ( b 3 - a 1 ) z - a 1 ( 1 + O ( 1 z ) ) + Γ ( a 2 ) Γ ( a 1 - a 2 ) Γ ( b 1 - a 2 ) Γ ( b 2 - a 2 ) Γ ( b 3 - a 2 ) z - a 2 ( 1 + O ( 1 z ) ) + ( - z ) χ π ( cos ( π χ + 2 - z ) ( 1 + O ( 1 z ) ) + 1 16 - z ( 2 ( 3 a 1 + 3 a 2 + b 1 + b 2 + b 3 - 2 ) ( 4 χ - 1 ) + 16 ( b 1 b 2 + b 1 b 3 + b 2 b 3 - a 1 a 2 ) - 3 ) sin ( π χ + 2 - z ) ( 1 + O ( 1 z ) ) ) ) /; χ 1 2 ( a 1 + a 2 - b 1 - b 2 - b 3 + 1 2 ) a 1 a 2 ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript b 1 Subscript b 2 Subscript b 3 -1 z Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript b 3 Gamma Subscript a 1 Gamma Subscript a 2 -1 Gamma Subscript a 1 Gamma Subscript a 2 -1 Subscript a 1 Gamma Subscript b 1 -1 Subscript a 1 Gamma Subscript b 2 -1 Subscript a 1 Gamma Subscript b 3 -1 Subscript a 1 -1 z -1 Subscript a 1 1 O 1 z -1 Gamma Subscript a 2 Gamma Subscript a 1 -1 Subscript a 2 Gamma Subscript b 1 -1 Subscript a 2 Gamma Subscript b 2 -1 Subscript a 2 Gamma Subscript b 3 -1 Subscript a 2 -1 z -1 Subscript a 2 1 O 1 z -1 -1 z χ 1 2 -1 χ 2 -1 z 1 2 1 O 1 z -1 1 16 -1 z 1 2 -1 2 3 Subscript a 1 3 Subscript a 2 Subscript b 1 Subscript b 2 Subscript b 3 -2 4 χ -1 16 Subscript b 1 Subscript b 2 Subscript b 1 Subscript b 3 Subscript b 2 Subscript b 3 -1 Subscript a 1 Subscript a 2 -3 χ 2 -1 z 1 2 1 O 1 z -1 χ 1 2 Subscript a 1 Subscript a 2 -1 Subscript b 1 -1 Subscript b 2 -1 Subscript b 3 1 2 Subscript a 1 Subscript a 2 Rule z [/itex]

 Rule Form

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

 2001-10-29