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 NorlundB

 http://functions.wolfram.com/05.17.06.0020.01

 Input Form

 NorlundB[n, \[Alpha], z] \[Proportional] z^n UnitStep[n] + \[Alpha] ((n/2) z^(n - 1) UnitStep[n - 1] + Sum[Binomial[n, k] z^k (n - k + 1 - 2 Floor[(n - k)/2]) (n - k)! Sum[(((-1)^j Binomial[n - k, j])/j) Subscript[p, j, n - k], {j, 1, n - k}], {k, 0, n}] - Sum[Binomial[n, k] z^k (n - k - 2 Floor[(n - k)/2]) (n - k)! Sum[(((-1)^j Binomial[n - k, j])/j) (1/j - HarmonicNumber[n - k]) Subscript[p, j, n - k] \[Alpha], {j, 1, n - k}], {k, 0, n}]) (1 + O[\[Alpha]]) /; 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", ",", "\[Alpha]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["UnitStep", "[", "n", "]"]]]], "+", RowBox[List["\[Alpha]", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", "n"]], "2"], SuperscriptBox["z", RowBox[List["n", "-", "1"]]], RowBox[List["UnitStep", "[", RowBox[List["n", "-", "1"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], SuperscriptBox["z", "k"], RowBox[List["(", RowBox[List["n", "-", "k", "+", "1", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "k"]], "2"], "]"]]]]]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "k"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "j"]], "]"]]]], "j"], SubscriptBox["p", RowBox[List["j", ",", RowBox[List["n", "-", "k"]]]]]]]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], SuperscriptBox["z", "k"], RowBox[List["(", RowBox[List["n", "-", "k", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "k"]], "2"], "]"]]]]]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "k"]]], " ", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "j"]], "]"]]]], "j"], RowBox[List["(", RowBox[List[FractionBox["1", "j"], "-", " ", RowBox[List["HarmonicNumber", "[", RowBox[List["n", "-", "k"]], "]"]]]], ")"]], SubscriptBox["p", RowBox[List["j", ",", RowBox[List["n", "-", "k"]]]]], "\[Alpha]"]]]]]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "\[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

 B TagBox["B", NorlundB] n ( α ) ( z ) θ UnitStep ( n ) z n + α ( n 2 θ UnitStep ( n - 1 ) z n - 1 + k = 0 n ( 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 k ( n - k - 2 n - k 2 + 1 ) ( n - k ) ! j = 1 n - k ( - 1 ) j j ( n - k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "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 , n - k - k = 0 n ( 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 k ( n - k - 2 n - k 2 ) ( n - k ) ! j = 1 n - k ( - 1 ) j j ( n - k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "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]] ( 1 j - H HarmonicNumber n - k ) p j , n - k α ) ( 1 + O ( 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 Proportional NorlundB n α z UnitStep n z n α n 2 -1 UnitStep n -1 z n -1 k 0 n Binomial n k z k n -1 k -1 2 n -1 k 2 -1 1 n -1 k j 1 n -1 k -1 j j -1 Binomial n -1 k j Subscript p j n -1 k -1 k 0 n Binomial n k z k n -1 k -1 2 n -1 k 2 -1 n -1 k j 1 n -1 k -1 j j -1 Binomial n -1 k j 1 j -1 -1 HarmonicNumber n -1 k Subscript p j n -1 k α 1 O 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["NorlundB", "[", RowBox[List["n_", ",", "\[Alpha]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["UnitStep", "[", "n", "]"]]]], "+", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "n", " ", SuperscriptBox["z", RowBox[List["n", "-", "1"]]], " ", RowBox[List["UnitStep", "[", RowBox[List["n", "-", "1"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"], " ", RowBox[List["(", RowBox[List["n", "-", "k", "+", "1", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "k"]], "2"], "]"]]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "k"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "j"]], "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["n", "-", "k"]]]]]]], "j"]]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"], " ", RowBox[List["(", RowBox[List["n", "-", "k", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "k"]], "2"], "]"]]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["n", "-", "k"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", "j"], "-", RowBox[List["HarmonicNumber", "[", RowBox[List["n", "-", "k"]], "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["n", "-", "k"]]]]], " ", "\[Alpha]"]], "j"]]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "\[Alpha]", "]"]]]], ")"]]]]]], "/;", 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