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 Beta

 http://functions.wolfram.com/06.20.21.0012.01

 Input Form

 Integrate[b^(\[Alpha] - 1) Beta[Subscript[z, 1], Subscript[z, 2], a, b], b] == Sum[(((-1)^k k! b^(k + \[Alpha]))/((a + j) (k + \[Alpha]) k! j!)) StirlingS1[j, k] (Subscript[z, 2]^(j + a) HypergeometricPFQ[ {1 + j, a + j}, {1 + a + j}, Subscript[z, 2]] - Subscript[z, 1]^(j + a) HypergeometricPFQ[{1 + j, a + j}, {1 + a + j}, Subscript[z, 1]]), {j, 0, Infinity}, {k, 0, j}] /; Abs[Subscript[z, 1]] < 1 && Abs[Subscript[z, 2]] < 1 && !(Element[-a, Integers] && -a >= 0)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["b", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], RowBox[List["\[DifferentialD]", "b"]]]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "j"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["k", "!"]], SuperscriptBox["b", RowBox[List["k", "+", "\[Alpha]"]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "j"]], ")"]], RowBox[List["(", RowBox[List["k", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["k", "!"]], RowBox[List["j", "!"]]]]], " ", RowBox[List["StirlingS1", "[", RowBox[List["j", ",", "k"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[SubsuperscriptBox["z", "2", RowBox[List["j", "+", "a"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "j"]], ",", RowBox[List["a", "+", "j"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", "a", "+", "j"]], "}"]], ",", SubscriptBox["z", "2"]]], "]"]]]], "-", RowBox[List[SubsuperscriptBox["z", "1", RowBox[List["j", "+", "a"]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "j"]], ",", RowBox[List["a", "+", "j"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", "a", "+", "j"]], "}"]], ",", SubscriptBox["z", "1"]]], "]"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", SubscriptBox["z", "1"], "]"]], "<", "1"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["z", "2"], "]"]], "<", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], "]"]]]]]]]]

 MathML Form

 b α - 1 Β TagBox["\[CapitalBeta]", BetaRegularized] ( z 1 , z 2 ) ( a , b ) b j = 0 k = 0 j ( - 1 ) k k ! b k + α S TagBox["S", StirlingS1] j ( k ) ( a + j ) ( k + α ) k ! j ! ( z 2 a + j 2 F 1 ( j + 1 , a + j ; a + j + 1 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "j"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - z 1 a + j 2 F 1 ( j + 1 , a + j ; a + j + 1 ; z 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "j"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "1"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) /; "\[LeftBracketingBar]" z 1 "\[RightBracketingBar]" < 1 "\[LeftBracketingBar]" z 2 "\[RightBracketingBar]" < 1 - a b α - 1 Β TagBox["\[CapitalBeta]", BetaRegularized] ( z 1 , z 2 ) ( a , b ) b j = 0 k = 0 j ( - 1 ) k k ! b k + α S TagBox["S", StirlingS1] j ( k ) ( a + j ) ( k + α ) k ! j ! ( z 2 a + j 2 F 1 ( j + 1 , a + j ; a + j + 1 ; z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "j"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "2"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - z 1 a + j 2 F 1 ( j + 1 , a + j ; a + j + 1 ; z 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "j"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "+", "j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[SubscriptBox["z", "1"], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) /; "\[LeftBracketingBar]" z 1 "\[RightBracketingBar]" < 1 "\[LeftBracketingBar]" z 2 "\[RightBracketingBar]" < 1 - a [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["b_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", "b_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "j"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["b", RowBox[List["k", "+", "\[Alpha]"]]]]], ")"]], " ", RowBox[List["StirlingS1", "[", RowBox[List["j", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SubsuperscriptBox["zz", "2", RowBox[List["j", "+", "a"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "j"]], ",", RowBox[List["a", "+", "j"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", "a", "+", "j"]], "}"]], ",", SubscriptBox["zz", "2"]]], "]"]]]], "-", RowBox[List[SubsuperscriptBox["zz", "1", RowBox[List["j", "+", "a"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "j"]], ",", RowBox[List["a", "+", "j"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "+", "a", "+", "j"]], "}"]], ",", SubscriptBox["zz", "1"]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "j"]], ")"]], " ", RowBox[List["(", RowBox[List["k", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["j", "!"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", SubscriptBox["zz", "1"], "]"]], "<", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["zz", "2"], "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]]

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

 2001-10-29