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; }

 Beta

 http://functions.wolfram.com/06.19.20.0010.01

 Input Form

 D[Beta[z, a, b], {b, n}] == (-1)^n n! Gamma[b]^(n + 1) HypergeometricPFQRegularized[{Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, n + 1], 1 - a}, {1 + Subscript[a, 1], 1 + Subscript[a, 2], \[Ellipsis], 1 + Subscript[a, n + 1]}, 1] - Gamma[b] (1 - z)^b Log[1 - z]^n Sum[Binomial[n, j] j! HypergeometricPFQRegularized[ {Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, j + 1], 1 - a}, {1 + Subscript[a, 1], 1 + Subscript[a, 2], \[Ellipsis], 1 + Subscript[a, j + 1]}, 1 - z] (-(Gamma[b]/Log[1 - z]))^j, {j, 0, n}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == b && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["b", ",", "n"]], "}"]]], RowBox[List["Beta", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "b", "]"]], RowBox[List["n", "+", "1"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", RowBox[List["1", "-", "a"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["n", "+", "1"]]]]]]], "}"]], ",", "1"]], "]"]]]], "-", RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "b"], SuperscriptBox[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["j", "!"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["j", "+", "1"]]], ",", RowBox[List["1", "-", "a"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["1", "-", "z"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["Gamma", "[", "b", "]"]], RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]]], ")"]], "j"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", "b"]], "\[And]", " ", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n Β Beta z ( a , b ) b n ( - 1 ) n n ! Γ ( b ) n + 1 n + 2 F ~ n + 1 ( a 1 , a 2 , , a n + 1 , 1 - a ; a 1 + 1 , a 2 + 1 , , a n + 1 + 1 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["n", "+", "2"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], RowBox[List["n", "+", "1"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "a"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] - Γ ( b ) ( 1 - z ) b log n ( 1 - z ) j = 0 n ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, True]] j ! j + 2 F ~ j + 1 ( a 1 , a 2 , , a j + 1 , 1 - a ; a 1 + 1 , a 2 + 1 , , a j + 1 + 1 ; 1 - z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["j", "+", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "a"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", RowBox[List["j", "+", "1"]]], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[RowBox[List["1", "-", "z"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] ( - Γ ( b ) log ( 1 - z ) ) j /; a 1 a 2 a n + 1 b n Condition b n Beta z a b -1 n n Gamma b n 1 HypergeometricPFQRegularized Subscript a 1 Subscript a 2 Subscript a n 1 1 -1 a Subscript a 1 1 Subscript a 2 1 Subscript a n 1 1 1 -1 Gamma b 1 -1 z b 1 -1 z n j 0 n Binomial n j j HypergeometricPFQRegularized Subscript a 1 Subscript a 2 Subscript a j 1 1 -1 a Subscript a 1 1 Subscript a 2 1 Subscript a j 1 1 1 -1 z -1 Gamma b 1 -1 z -1 j Subscript a 1 Subscript a 2 Subscript a n 1 b n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["b_", ",", "n_"]], "}"]]]]], RowBox[List["Beta", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "b", "]"]], RowBox[List["n", "+", "1"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", RowBox[List["1", "-", "a"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["n", "+", "1"]]]]]]], "}"]], ",", "1"]], "]"]]]], "-", RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "b"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["j", "+", "1"]]], ",", RowBox[List["1", "-", "a"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["1", "-", "z"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["Gamma", "[", "b", "]"]], RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]]], ")"]], "j"]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["aa", "1"], "\[Equal]", SubscriptBox["aa", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", "b"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29