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/07.27.06.0010.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, 2}]/Product[Gamma[Subscript[a, k]], {k, 1, 3}]) ((1 - z)^Subscript[\[Psi], 2] Sum[Subscript[h, j] (1 - j)^j, {j, 0, -Subscript[\[Psi], 2] - 1}] + Sum[(Subscript[u, j] + Subscript[v, j] Log[1 - z]) (1 - z)^j, {j, 0, Infinity}]) /; Abs[1 - z] < 1 && Subscript[\[Psi], 2] == Sum[Subscript[b, j], {j, 1, 2}] - Sum[Subscript[a, j], {j, 1, 3}] && Subscript[h, j] == ((1/j!) ((-1)^j Pochhammer[Subscript[a, 1] + Subscript[\[Psi], 2], j] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], 2], j] (-Subscript[\[Psi], 2] - j - 1)!)) HypergeometricPFQRegularized[ {Subscript[b, 1] - Subscript[a, 3], Subscript[b, 2] - Subscript[a, 3], -j}, {Subscript[a, 1] + Subscript[\[Psi], 2], Subscript[a, 2] + Subscript[\[Psi], 2]}, 1] && (Subscript[u, j] == ((Pochhammer[Subscript[a, 1] + Subscript[\[Psi], 2], j - Subscript[\[Psi], 2]] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], 2], j - Subscript[\[Psi], 2]])/ (j! (j - Subscript[\[Psi], 2])!)) ((-1)^Subscript[\[Psi], 2] Sum[((Pochhammer[Subscript[b, 1] - Subscript[a, 3], k] Pochhammer[Subscript[b, 2] - Subscript[a, 3], k] Pochhammer[Subscript[\[Psi], 2] - j, k])/ (Pochhammer[Subscript[a, 1] + Subscript[\[Psi], 2], k] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], 2], k] k!)) (PolyGamma[j + 1] + PolyGamma[1 - Subscript[\[Psi], 2] + j - k] - PolyGamma[Subscript[a, 1] + j] - PolyGamma[Subscript[a, 2] + j]), {k, 0, j - Subscript[\[Psi], 2]}] + (-1)^j (j - Subscript[\[Psi], 2])! Sum[(Pochhammer[Subscript[b, 1] - Subscript[a, 3], k] Pochhammer[Subscript[b, 2] - Subscript[a, 3], k] (k + Subscript[\[Psi], 2] - j - 1)!)/ (Pochhammer[Subscript[a, 1] + Subscript[\[Psi], 2], k] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], 2], k] k!), {k, j - Subscript[\[Psi], 2] + 1, Infinity}]) /; Re[Subscript[a, 3]] > -j) && Subscript[v, j] == ((1/(j! (j - Subscript[\[Psi], 2])!)) ((-1)^(Subscript[\[Psi], 2] - 1) Pochhammer[Subscript[a, 1] + Subscript[\[Psi], 2], j - Subscript[\[Psi], 2]] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], 2], j - Subscript[\[Psi], 2]])) HypergeometricPFQ[{Subscript[b, 1] - Subscript[a, 3], Subscript[b, 2] - Subscript[a, 3], Subscript[\[Psi], 2] - j}, {Subscript[a, 1] + Subscript[\[Psi], 2], Subscript[a, 2] + Subscript[\[Psi], 2]}, 1] && Element[-Subscript[\[Psi], 2], Integers] && -Subscript[\[Psi], 2] >= 0

 Standard Form

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 MathML Form

 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] k = 1 2 Γ ( b k ) k = 1 3 Γ ( a k ) ( ( 1 - z ) ψ 2 j = 0 - ψ 2 - 1 h j ( 1 - j ) j + j = 0 ( u j + v j log ( 1 - z ) ) ( 1 - z ) j ) /; "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 1 ψ 2 j = 1 2 b j - j = 1 3 a j h j ( - 1 ) j ( a 1 + ψ 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "j"], Pochhammer] ( a 2 + ψ 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "j"], Pochhammer] ( - ψ 2 - j - 1 ) ! j ! 3 F ~ 2 ( b 1 - a 3 , b 2 - a 3 , - j ; a 1 + ψ 2 , a 2 + ψ 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "j"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ( u j ( a 1 + ψ 2 ) j - ψ 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], RowBox[List["j", "-", SubscriptBox["\[Psi]", "2"]]]], Pochhammer] ( a 2 + ψ 2 ) j - ψ 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], RowBox[List["j", "-", SubscriptBox["\[Psi]", "2"]]]], Pochhammer] j ! ( j - ψ 2 ) ! ( ( - 1 ) j ( j - ψ 2 ) ! k = j - ψ 2 + 1 ( b 1 - a 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], ")"]], "k"], Pochhammer] ( b 2 - a 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], ")"]], "k"], Pochhammer] ( k - j + ψ 2 - 1 ) ! ( a 1 + ψ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "k"], Pochhammer] ( a 2 + ψ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "k"], Pochhammer] k ! + ( - 1 ) ψ 2 k = 0 j - ψ 2 ( b 1 - a 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], ")"]], "k"], Pochhammer] ( b 2 - a 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], ")"]], "k"], Pochhammer] ( ψ 2 - j ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["\[Psi]", "2"], "-", "j"]], ")"]], "k"], Pochhammer] ( a 1 + ψ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "k"], Pochhammer] ( a 2 + ψ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "k"], Pochhammer] k ! ( ψ TagBox["\[Psi]", PolyGamma] ( j + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + a 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + a 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( j - k - ψ 2 + 1 ) ) ) /; Re ( a 3 ) > - j ) v j ( - 1 ) ψ 2 - 1 ( a 1 + ψ 2 ) j - ψ 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], RowBox[List["j", "-", SubscriptBox["\[Psi]", "2"]]]], Pochhammer] ( a 2 + ψ 2 ) j - ψ 2 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], RowBox[List["j", "-", SubscriptBox["\[Psi]", "2"]]]], Pochhammer] j ! ( j - ψ 2 ) ! 3 F 2 ( b 1 - a 3 , b 2 - a 3 , ψ 2 - j ; a 1 + ψ 2 , a 2 + ψ 2 ; 1 ) 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[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["\[Psi]", "2"], "-", "j"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - ψ 2 Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript b 1 Subscript b 2 z k 1 2 Gamma Subscript b k k 1 3 Gamma Subscript a k -1 1 -1 z Subscript ψ 2 j 0 -1 Subscript ψ 2 -1 Subscript h j 1 -1 j j j 0 Subscript u j Subscript v j 1 -1 z 1 -1 z j 1 -1 z 1 Subscript ψ 2 j 1 2 Subscript b j -1 j 1 3 Subscript a j Subscript h j -1 j Pochhammer Subscript a 1 Subscript ψ 2 j Pochhammer Subscript a 2 Subscript ψ 2 j -1 Subscript ψ 2 -1 j -1 j -1 HypergeometricPFQRegularized Subscript b 1 -1 Subscript a 3 Subscript b 2 -1 Subscript a 3 -1 j Subscript a 1 Subscript ψ 2 Subscript a 2 Subscript ψ 2 1 Condition Subscript u j Pochhammer Subscript a 1 Subscript ψ 2 j -1 Subscript ψ 2 Pochhammer Subscript a 2 Subscript ψ 2 j -1 Subscript ψ 2 j j -1 Subscript ψ 2 -1 -1 j j -1 Subscript ψ 2 k j -1 Subscript ψ 2 1 Pochhammer Subscript b 1 -1 Subscript a 3 k Pochhammer Subscript b 2 -1 Subscript a 3 k k -1 j Subscript ψ 2 -1 Pochhammer Subscript a 1 Subscript ψ 2 k Pochhammer Subscript a 2 Subscript ψ 2 k k -1 -1 Subscript ψ 2 k 0 j -1 Subscript ψ 2 Pochhammer Subscript b 1 -1 Subscript a 3 k Pochhammer Subscript b 2 -1 Subscript a 3 k Pochhammer Subscript ψ 2 -1 j k Pochhammer Subscript a 1 Subscript ψ 2 k Pochhammer Subscript a 2 Subscript ψ 2 k k -1 PolyGamma j 1 -1 PolyGamma j Subscript a 1 -1 PolyGamma j Subscript a 2 PolyGamma j -1 k -1 Subscript ψ 2 1 Subscript a 3 -1 j Subscript v j -1 Subscript ψ 2 -1 Pochhammer Subscript a 1 Subscript ψ 2 j -1 Subscript ψ 2 Pochhammer Subscript a 2 Subscript ψ 2 j -1 Subscript ψ 2 j j -1 Subscript ψ 2 -1 HypergeometricPFQ Subscript b 1 -1 Subscript a 3 Subscript b 2 -1 Subscript a 3 Subscript ψ 2 -1 j Subscript a 1 Subscript ψ 2 Subscript a 2 Subscript ψ 2 1 -1 Subscript ψ 2 [/itex]

 Rule Form

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

 2001-10-29