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 NorlundB

 http://functions.wolfram.com/05.16.20.0006.01

 Input Form

 D[NorlundB[n, z], {z, \[Alpha]}] == KroneckerDelta[n]/(z^\[Alpha] Gamma[1 - \[Alpha]]) - (-1)^n Sum[(i!/Gamma[i - \[Alpha] + 1]) Sum[StirlingS1[n + 1, k] Sum[(-1)^(j + i) j^(k - i - 1) Binomial[n, j] Subscript[p, j, n] z^(i - \[Alpha]), {j, 1, n}], {k, 1, i}], {i, 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["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["NorlundB", "[", RowBox[List["n", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List["i", "!"]], RowBox[List["Gamma", "[", RowBox[List["i", "-", "\[Alpha]", "+", "1"]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "i"], " ", RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "k"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "i"]]], " ", SuperscriptBox["j", RowBox[List["k", "-", "i", "-", "1"]]], RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], SubscriptBox["p", RowBox[List["j", ",", "n"]]], SuperscriptBox["z", RowBox[List["i", "-", "\[Alpha]"]]]]]]]]]]]]]]]]]]]]], "/;", 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

 α B TagBox["B", NorlundB] n ( z ) z α δ KroneckerDelta n z - α Γ ( 1 - α ) - ( - 1 ) n i = 0 n i ! Γ ( i - α + 1 ) k = 1 i S TagBox["S", StirlingS1] n + 1 ( k ) j = 1 n ( - 1 ) i + j j k - i - 1 ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", 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 , n z i - α /; 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 z α NorlundB n z KroneckerDelta n z -1 α Gamma 1 -1 α -1 -1 -1 n i 0 n i Gamma i -1 α 1 -1 k 1 i StirlingS1 n 1 k j 1 n -1 i j j k -1 i -1 Binomial n j Subscript p j n z i -1 α 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["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["NorlundB", "[", RowBox[List["n_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "i"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "i"]]], " ", SuperscriptBox["j", RowBox[List["k", "-", "i", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", "n"]]], " ", SuperscriptBox["z", RowBox[List["i", "-", "\[Alpha]"]]]]]]]]]]]]], RowBox[List["Gamma", "[", RowBox[List["i", "-", "\[Alpha]", "+", "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