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/07.26.06.0011.01

 Input Form

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

 Standard Form

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 MathML Form

 2 F 3 ( a 1 , n + a 1 ; 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[RowBox[List["n", "+", SubscriptBox["a", "1"]]], 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 ) Γ ( n + a 1 ) Γ ( b 1 - a 1 ) Γ ( b 2 - a 1 ) Γ ( b 3 - a 1 ) ( - z ) - n - a 1 k = 0 ( a 1 ) k + n TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], RowBox[List["k", "+", "n"]]], Pochhammer] ( a 1 - b 1 + 1 ) k + n TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"], "+", "1"]], ")"]], RowBox[List["k", "+", "n"]]], Pochhammer] ( a 1 - b 2 + 1 ) k + n TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"], "+", "1"]], ")"]], RowBox[List["k", "+", "n"]]], Pochhammer] ( a 1 - b 3 + 1 ) k + n TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["b", "3"], "+", "1"]], ")"]], RowBox[List["k", "+", "n"]]], Pochhammer] k ! ( k + n ) ! ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + n + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + n + a 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( b 1 - a 1 - n - k ) - ψ TagBox["\[Psi]", PolyGamma] ( b 2 - a 1 - n - k ) - ψ TagBox["\[Psi]", PolyGamma] ( b 3 - a 1 - n - k ) ) z - k + Γ ( b 1 ) Γ ( b 2 ) Γ ( b 3 ) 2 π Γ ( a 1 ) Γ ( n + a 1 ) ( - 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 ) n ! Γ ( a 1 ) Γ ( b 1 - a 1 - n ) Γ ( b 2 - a 1 - n ) Γ ( b 3 - a 1 - n ) z - n ( - z ) - a 1 log ( - z ) 4 F 1 ( n + a 1 , n + a 1 - b 1 + 1 , n + a 1 - b 2 + 1 , n + a 1 - b 3 + 1 ; n + 1 ; 1 z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["4", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["n", "+", SubscriptBox["a", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["n", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "3"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[FractionBox["1", "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 ( n + a 1 ) ( - z ) - a 1 k = 0 n - 1 ( ( a 1 ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], "k"], Pochhammer] Γ ( n - k ) ) z - k Γ ( - k - a 1 + b 1 ) Γ ( - k - a 1 + b 2 ) Γ ( - k - a 1 + b 3 ) k ! /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n χ 1 2 ( n + 2 a 1 - b 1 - b 2 - b 3 + 1 2 ) c 0 1 c 1 2 ( 𝔅 - 𝔄 + 1 4 ( 3 A 2 + B 3 - 2 ) ( A 2 - B 3 ) - 3 16 ) c 2 c 1 2 2 + 1 16 ( - 16 ( 2 A 2 - 3 ) ( 𝔅 - 𝔄 ) + 32 + 4 ( - 8 A 2 2 + 11 A 2 + 8 𝔄 + B 3 - 2 ) ( A 2 - B 3 ) - 3 ) c k 1 2 k ( ( k - 2 χ - 3 ) ( k - 2 χ - 2 b 1 - 1 ) ( k - 2 χ - 2 b 2 - 1 ) ( k - 2 χ - 2 b 3 - 1 ) c k - 3 - ( 4 ( k - 1 ) 3 - 6 ( 4 χ + B 3 ) ( k - 1 ) 2 + 2 ( 24 χ 2 + 12 B 3 χ + 4 𝔅 + B 3 - 1 ) ( k - 1 ) - 32 χ 3 - 24 B 3 χ 2 - 4 𝔅 - 8 - 4 ( 4 𝔅 + B 3 - 1 ) χ + 2 B 3 - 1 ) c k - 2 + ( 5 ( k - 1 ) 2 + 2 ( - 10 χ + A 2 - 3 B 3 + 3 ) ( k - 1 ) + 2 c 1 ) c k - 1 ) A 2 n + 2 a 1 B 3 b 1 + b 2 + b 3 𝔄 a 1 ( n + a 1 ) 𝔅 b 1 b 2 + b 3 b 2 + b 1 b 3 b 1 b 2 b 3 Condition Proportional HypergeometricPFQ Subscript a 1 n Subscript a 1 Subscript b 1 Subscript b 2 Subscript b 3 z Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript b 3 Gamma n 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 -1 z -1 n -1 Subscript a 1 k 0 Pochhammer Subscript a 1 k n Pochhammer Subscript a 1 -1 Subscript b 1 1 k n Pochhammer Subscript a 1 -1 Subscript b 2 1 k n Pochhammer Subscript a 1 -1 Subscript b 3 1 k n k k n -1 PolyGamma k 1 PolyGamma k n 1 -1 PolyGamma k n Subscript a 1 -1 PolyGamma Subscript b 1 -1 Subscript a 1 -1 n -1 k -1 PolyGamma Subscript b 2 -1 Subscript a 1 -1 n -1 k -1 PolyGamma Subscript b 3 -1 Subscript a 1 -1 n -1 k z -1 k Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript b 3 2 1 2 Gamma Subscript a 1 Gamma n Subscript a 1 -1 -1 z χ χ 2 -1 z 1 2 k 0 -1 k 2 -1 k Subscript c k -1 z -1 k 2 -1 -1 χ 2 -1 z 1 2 k 0 k 2 -1 k Subscript c k -1 z -1 k 2 -1 Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript b 3 n Gamma Subscript a 1 Gamma Subscript b 1 -1 Subscript a 1 -1 n Gamma Subscript b 2 -1 Subscript a 1 -1 n Gamma Subscript b 3 -1 Subscript a 1 -1 n -1 z -1 n -1 z -1 Subscript a 1 -1 z HypergeometricPFQ n Subscript a 1 n Subscript a 1 -1 Subscript b 1 1 n Subscript a 1 -1 Subscript b 2 1 n Subscript a 1 -1 Subscript b 3 1 n 1 1 z -1 Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript b 3 Gamma n Subscript a 1 -1 -1 z -1 Subscript a 1 k 0 n -1 Pochhammer Subscript a 1 k Gamma n -1 k z -1 k Gamma -1 k -1 Subscript a 1 Subscript b 1 Gamma -1 k -1 Subscript a 1 Subscript b 2 Gamma -1 k -1 Subscript a 1 Subscript b 3 k -1 Rule z n χ 1 2 n 2 Subscript a 1 -1 Subscript b 1 -1 Subscript b 2 -1 Subscript b 3 1 2 Subscript c 0 1 Subscript c 1 2 𝔅 -1 𝔄 1 4 3 Subscript A 2 Subscript B 3 -2 Subscript A 2 -1 Subscript B 3 -1 3 16 Subscript c 2 Subscript c 1 2 2 -1 1 16 -16 2 Subscript A 2 -3 𝔅 -1 𝔄 32 4 -8 Subscript A 2 2 11 Subscript A 2 8 𝔄 Subscript B 3 -2 Subscript A 2 -1 Subscript B 3 -3 Subscript c k 1 2 k -1 k -1 2 χ -3 k -1 2 χ -1 2 Subscript b 1 -1 k -1 2 χ -1 2 Subscript b 2 -1 k -1 2 χ -1 2 Subscript b 3 -1 Subscript c k -3 -1 4 k -1 3 -1 6 4 χ Subscript B 3 k -1 2 2 24 χ 2 12 Subscript B 3 χ 4 𝔅 Subscript B 3 -1 k -1 -1 32 χ 3 -1 24 Subscript B 3 χ 2 -1 4 𝔅 -1 8 -1 4 4 𝔅 Subscript B 3 -1 χ 2 Subscript B 3 -1 Subscript c k -2 5 k -1 2 2 -10 χ Subscript A 2 -1 3 Subscript B 3 3 k -1 2 Subscript c 1 Subscript c k -1 Subscript A 2 n 2 Subscript a 1 Subscript B 3 Subscript b 1 Subscript b 2 Subscript b 3 𝔄 Subscript a 1 n Subscript a 1 𝔅 Subscript b 1 Subscript b 2 Subscript b 3 Subscript b 2 Subscript b 1 Subscript b 3 Subscript b 1 Subscript b 2 Subscript b 3 [/itex]

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

 2001-10-29