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 NorlundB

 http://functions.wolfram.com/05.16.06.0008.01

 Input Form

 NorlundB[n, z] \[Proportional] KroneckerDelta[n] + ((-1)^n z^n (1 + 2^n Sum[StirlingS1[1 + n, k] Sum[(-1)^(j + n) j^(k - n) Binomial[n, j] Subscript[p, j, n] (1/z), {j, 1, n}], {k, 1, -1 + n}] - 2^n Sum[StirlingS1[1 + n, k] Sum[(-1)^(j + n) j^(1 + k - n) Binomial[n, j] Subscript[p, j, n] (1/z^2), {j, 1, n}], {k, 1, -2 + n}] + \[Ellipsis]))/2^n /; (Abs[z] -> Infinity) && 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 && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["NorlundB", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], SuperscriptBox["2", RowBox[List["-", "n"]]], SuperscriptBox["z", "n"], RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "n"]]], " ", SuperscriptBox["j", RowBox[List["k", "-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], SubscriptBox["p", RowBox[List["j", ",", "n"]]], FractionBox["1", "z"]]]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["-", "2"]], "+", "n"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "n"]]], " ", SuperscriptBox["j", RowBox[List["1", "+", "k", "-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], SubscriptBox["p", RowBox[List["j", ",", "n"]]], FractionBox["1", SuperscriptBox["z", "2"]]]]]]]]]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "->", "\[Infinity]"]], ")"]], "\[And]", 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"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 B TagBox["B", NorlundB] n ( z ) δ KroneckerDelta n + ( - 1 ) n 2 - n z n ( 1 + 2 n k = 1 n - 1 S TagBox["S", StirlingS1] n + 1 ( k ) j = 1 n ( - 1 ) j + n j k - n ( 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 1 z - 2 n k = 1 n - 2 S TagBox["S", StirlingS1] n + 1 ( k ) j = 1 n ( - 1 ) j + n j k - n + 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 1 z 2 + ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) 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]] n > 0 Condition Proportional NorlundB n z KroneckerDelta n -1 n 2 -1 n z n 1 2 n k 1 n -1 StirlingS1 n 1 k j 1 n -1 j n j k -1 n Binomial n j Subscript p j n 1 z -1 -1 2 n k 1 n -2 StirlingS1 n 1 k j 1 n -1 j n j k -1 n 1 Binomial n j Subscript p j n 1 z 2 -1 Rule z 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 n 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["NorlundB", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["-", "n"]]], " ", SuperscriptBox["z", "n"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "n"]]], " ", SuperscriptBox["j", RowBox[List["k", "-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", "n"]]]]], "z"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["-", "2"]], "+", "n"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "n"]]], " ", SuperscriptBox["j", RowBox[List["1", "+", "k", "-", "n"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", "n"]]]]], SuperscriptBox["z", "2"]]]]]]]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", 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"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02