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 NorlundB

 http://functions.wolfram.com/05.17.20.0007.01

 Input Form

 D[NorlundB[n, \[Alpha], z], {\[Alpha], \[Beta]}] == (z^n UnitStep[n])/(\[Alpha]^\[Beta] Gamma[1 - \[Beta]]) - Sum[(-1)^k Binomial[n, k] z^(n - k) Sum[(i!/Gamma[i - \[Beta] + 1]) Sum[StirlingS1[k + 1, r] Sum[(-1)^(j + i) j^(r - i - 1) Binomial[k, j] p[j, k] \[Alpha]^(i - \[Beta]), {j, 1, k}], {r, 1, i}], {i, 0, k}], {k, 0, n}] /; Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/k) Sum[(j m - k + m) Subscript[a, m] Subscript[p, j, k - m], {m, 1, k}] && Subscript[a, k] == 1/(k + 1)! && Element[k, Integers] && k >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Alpha]", ",", "\[Beta]"]], "}"]]], RowBox[List["NorlundB", "[", RowBox[List["n", ",", "\[Alpha]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox["\[Alpha]", RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["UnitStep", "[", "n", "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Beta]"]], "]"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", " ", "k"]], "]"]], SuperscriptBox["z", RowBox[List["n", "-", "k"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List["i", "!"]], RowBox[List["Gamma", "[", RowBox[List["i", "-", "\[Beta]", "+", "1"]], "]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "i"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["k", "+", "1"]], ",", "r"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "i"]]], " ", SuperscriptBox["j", RowBox[List["r", "-", "i", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["p", "[", RowBox[List["j", ",", "k"]], "]"]], " ", SuperscriptBox["\[Alpha]", RowBox[List["i", "-", "\[Beta]"]]]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List[FractionBox["1", "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], SubscriptBox["a", "m"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]]]]]], "\[And]", RowBox[List[SubscriptBox["a", "k"], "\[Equal]", FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 β B TagBox["B", NorlundB] n ( α ) ( z ) α β z n α - β θ UnitStep ( n ) Γ ( 1 - β ) - k = 0 n ( - 1 ) k ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] z n - k i = 0 k i ! Γ ( i - β + 1 ) r = 1 i S TagBox["S", StirlingS1] k + 1 ( r ) j = 1 k ( - 1 ) i + j j r - i - 1 ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["j", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] p ( j , k ) α i - β /; p j , 0 1 p j , k 1 k m = 1 k ( j m + m - k ) a m p j , k - m a k 1 ( k + 1 ) ! k TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition α β NorlundB n α z z n α -1 β UnitStep n Gamma 1 -1 β -1 -1 k 0 n -1 k Binomial n k z n -1 k i 0 k i Gamma i -1 β 1 -1 r 1 i StirlingS1 k 1 r j 1 k -1 i j j r -1 i -1 Binomial k j p j k α i -1 β Subscript p j 0 1 Subscript p j k 1 k -1 m 1 k j m m -1 k Subscript a m Subscript p j k -1 m Subscript a k 1 k 1 -1 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Alpha]_", ",", "\[Beta]_"]], "}"]]]]], RowBox[List["NorlundB", "[", RowBox[List["n_", ",", "\[Alpha]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox["\[Alpha]", RowBox[List["-", "\[Beta]"]]], " ", RowBox[List["UnitStep", "[", "n", "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Beta]"]], "]"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["n", "-", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "i"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["k", "+", "1"]], ",", "r"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "i"]]], " ", SuperscriptBox["j", RowBox[List["r", "-", "i", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["p", "[", RowBox[List["j", ",", "k"]], "]"]], " ", SuperscriptBox["\[Alpha]", RowBox[List["i", "-", "\[Beta]"]]]]]]]]]]]]], RowBox[List["Gamma", "[", RowBox[List["i", "-", "\[Beta]", "+", "1"]], "]"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], " ", SubscriptBox["a", "m"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]], "k"]]], "&&", RowBox[List[SubscriptBox["a", "k"], "\[Equal]", FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02

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