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variants of this functions
Beta




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Beta[z1,z2,a,b] > Differentiation > Fractional integro-differentiation > With respect to b





http://functions.wolfram.com/06.20.20.0020.01









  


  








Input Form





D[Beta[Subscript[z, 1], Subscript[z, 2], a, b], {b, \[Alpha]}] == Sum[(((-1)^k k! b^(k - \[Alpha]))/((a + j) Gamma[k - \[Alpha] + 1] 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





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["b_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["Beta", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "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["Gamma", "[", RowBox[List["k", "-", "\[Alpha]", "+", "1"]], "]"]], " ", 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