html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.26.06.0009.01

 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]])/ (2 Sqrt[Pi] Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]])) (-z)^\[Chi] (E^(I (\[Chi] Pi + 2 Sqrt[-z])) Sum[((-I)^k Subscript[c, k])/ (2^k (-z)^(k/2)), {k, 0, Infinity}] + Sum[(I^k Subscript[c, k])/(2^k (-z)^(k/2)), {k, 0, Infinity}]/ E^(I (\[Chi] Pi + 2 Sqrt[-z]))) + (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]] Gamma[Subscript[a, 2] - Subscript[a, 1]])/(Gamma[Subscript[a, 2]] Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]] Gamma[Subscript[b, 3] - Subscript[a, 1]])) HypergeometricPFQ[{Subscript[a, 1], 1 + Subscript[a, 1] - Subscript[b, 1], 1 + Subscript[a, 1] - Subscript[b, 2], 1 + Subscript[a, 1] - Subscript[b, 3]}, {1 + Subscript[a, 1] - Subscript[a, 2]}, 1/z])/(-z)^Subscript[a, 1] + (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]] Gamma[Subscript[a, 1] - Subscript[a, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[b, 1] - Subscript[a, 2]] Gamma[Subscript[b, 2] - Subscript[a, 2]] Gamma[Subscript[b, 3] - Subscript[a, 2]])) HypergeometricPFQ[{Subscript[a, 2], 1 + Subscript[a, 2] - Subscript[b, 1], 1 + Subscript[a, 2] - Subscript[b, 2], 1 + Subscript[a, 2] - Subscript[b, 3]}, {1 + Subscript[a, 2] - Subscript[a, 1]}, 1/z])/(-z)^Subscript[a, 2] /; (Abs[z] -> Infinity) && \[Chi] == (1/2) (1/2 + Subscript[a, 1] + Subscript[a, 2] - Subscript[b, 1] - Subscript[b, 2] - Subscript[b, 3]) && Subscript[c, 0] == 1 && Subscript[c, 1] == 2 (\[GothicCapitalB] - \[GothicCapitalA] + (1/4) (Subscript[A, 2] - Subscript[B, 3]) (3 Subscript[A, 2] + Subscript[B, 3] - 2) - 3/16) && Subscript[c, 2] == Subscript[c, 1]^2/2 + (1/16) (32 \[GothicCapitalR] + 4 (Subscript[A, 2] - Subscript[B, 3]) (8 \[GothicCapitalA] + 11 Subscript[A, 2] - 8 Subscript[A, 2]^2 + Subscript[B, 3] - 2) - 16 (2 Subscript[A, 2] - 3) (\[GothicCapitalB] - \[GothicCapitalA]) - 3) && Subscript[c, k] == (1/(2 k)) ((2 (3 + Subscript[A, 2] - 3 Subscript[B, 3] - 10 \[Chi]) (k - 1) + 5 (k - 1)^2 + 2 Subscript[c, 1]) Subscript[c, k - 1] - (2 Subscript[B, 3] - 4 \[GothicCapitalB] - 8 \[GothicCapitalR] - 4 (Subscript[B, 3] + 4 \[GothicCapitalB] - 1) \[Chi] - 24 Subscript[B, 3] \[Chi]^2 - 32 \[Chi]^3 + 2 (Subscript[B, 3] + 4 \[GothicCapitalB] + 12 Subscript[B, 3] \[Chi] + 24 \[Chi]^2 - 1) (k - 1) - 6 (Subscript[B, 3] + 4 \[Chi]) (k - 1)^2 + 4 (k - 1)^3 - 1) Subscript[c, k - 2] + (k - 2 \[Chi] - 3) (k - 2 \[Chi] - 2 Subscript[b, 1] - 1) (k - 2 \[Chi] - 2 Subscript[b, 2] - 1) (k - 2 \[Chi] - 2 Subscript[b, 3] - 1) Subscript[c, k - 3]) && Subscript[A, 2] == Subscript[a, 1] + Subscript[a, 2] && Subscript[B, 3] == Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3] && \[GothicCapitalA] == Subscript[a, 1] Subscript[a, 2] && \[GothicCapitalB] == Subscript[b, 1] Subscript[b, 2] + Subscript[b, 1] Subscript[b, 3] + Subscript[b, 2] Subscript[b, 3] && \[GothicCapitalR] == Subscript[b, 1] Subscript[b, 2] Subscript[b, 3] && !Element[Subscript[a, 1] - Subscript[a, 2], Integers]

 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] Γ ( b 1 ) Γ ( b 2 ) Γ ( b 3 ) 2 π Γ ( a 1 ) Γ ( a 2 ) ( - z ) χ ( ( π χ + 2 - z ) k = 0 ( - ) k 2 - k c k ( - z ) - k 2 + - ( π χ + 2 - z ) k = 0 k 2 - k c k ( - z ) - k 2 ) + Γ ( b 1 ) Γ ( b 2 ) Γ ( b 3 ) Γ ( a 2 - a 1 ) Γ ( a 2 ) Γ ( b 1 - a 1 ) Γ ( b 2 - a 1 ) Γ ( b 3 - a 1 ) ( - z ) - a 1 4 F 1 ( a 1 , a 1 - b 1 + 1 , a 1 - b 2 + 1 , a 1 - b 3 + 1 ; 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 Date Added to functions.wolfram.com (modification date)

 2001-10-29