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 NorlundB

 http://functions.wolfram.com/05.16.06.0007.01

 Input Form

 NorlundB[n, z] \[Proportional] KroneckerDelta[n] + z ((1/2) KroneckerDelta[n - 1] + (n + 1 - 2 Floor[n/2]) n! Sum[(((-1)^j Binomial[n, j])/j) Subscript[p, j, n], {j, 1, n}] - (n - 2 Floor[n/2]) n! Sum[(((-1)^j Binomial[n, j])/j) (1/j - HarmonicNumber[n]) Subscript[p, j, n] z, {j, 1, n}]) (1 + O[z]) /; 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["NorlundB", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], "+", RowBox[List["z", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]], RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]]]], "j"], SubscriptBox["p", RowBox[List["j", ",", "n"]]]]]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]], RowBox[List["n", "!"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], " ", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]]]], "j"], RowBox[List["(", RowBox[List[FractionBox["1", "j"], "-", " ", RowBox[List["HarmonicNumber", "[", "n", "]"]]]], ")"]], SubscriptBox["p", RowBox[List["j", ",", "n"]]], "z"]]]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "z", "]"]]]], ")"]]]]]]]], "/;", 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

 B TagBox["B", NorlundB] n ( z ) δ KroneckerDelta n + z ( δ KroneckerDelta n - 1 2 + ( n - 2 n 2 + 1 ) n ! j = 1 n ( - 1 ) j j ( 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 - ( n - 2 n 2 ) n ! j = 1 n ( - 1 ) j j ( 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]] ( 1 j - H HarmonicNumber n ) p j , n z ) ( 1 + O ( z ) ) /; 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 Proportional NorlundB n z KroneckerDelta n z KroneckerDelta n -1 2 -1 n -1 2 n 2 -1 1 n j 1 n -1 j j -1 Binomial n j Subscript p j n -1 n -1 2 n 2 -1 n j 1 n -1 j j -1 Binomial n j 1 j -1 -1 HarmonicNumber n Subscript p j n z 1 O 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 [/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["z", " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["j", ",", "n"]]]]], "j"]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]], " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", "j"], "-", RowBox[List["HarmonicNumber", "[", "n", "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["j", ",", "n"]]], " ", "z"]], "j"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "z", "]"]]]], ")"]]]]]], "/;", 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