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 NorlundB

 http://functions.wolfram.com/05.17.06.0022.01

 Input Form

 NorlundB[n, \[Alpha], z] \[Proportional] z^n UnitStep[n] + z^n Sum[(Binomial[n, k] \[Alpha]^k (1 + 2^k Sum[StirlingS1[1 + k, r] Sum[(-1)^(j + k) j^(r - k) Binomial[k, j] Subscript[p, j, k] (1/\[Alpha]), {j, 1, k}], {r, 1, -1 + k}] - 2^k Sum[StirlingS1[1 + k, r] Sum[(-1)^(j + k) j^(1 + r - k) Binomial[k, j] Subscript[p, j, k] (1/\[Alpha]^2), {j, 1, k}], {r, 1, -2 + k}] + \[Ellipsis]))/ (-2 z)^k, {k, 0, n}] /; (Abs[\[Alpha]] -> 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", ",", "\[Alpha]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["UnitStep", "[", "n", "]"]]]], "+", RowBox[List[SuperscriptBox["z", "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "z"]], ")"]], RowBox[List["-", "k"]]], SuperscriptBox["\[Alpha]", "k"], RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["2", "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "k"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", "r"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["j", RowBox[List["r", "-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], SubscriptBox["p", RowBox[List["j", ",", "k"]]], FractionBox["1", "\[Alpha]"]]]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], RowBox[List[RowBox[List["-", "2"]], "+", "k"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", "r"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["j", RowBox[List["1", "+", "r", "-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], SubscriptBox["p", RowBox[List["j", ",", "k"]]], FractionBox["1", SuperscriptBox["\[Alpha]", "2"]]]]]]]]]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Alpha]", "]"]], "->", "\[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 ( α ) ( z ) θ UnitStep ( n ) z n + z n 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]] ( - 2 z ) - k α k ( 1 + 2 k r = 1 k - 1 S TagBox["S", StirlingS1] k + 1 ( r ) j = 1 k ( - 1 ) j + k j r - k α ( 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 - 2 k r = 1 k - 2 S TagBox["S", StirlingS1] k + 1 ( r ) j = 1 k ( - 1 ) j + k j r - k + 1 α 2 ( 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 + ) /; ( "\[LeftBracketingBar]" α "\[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 n > 0 Condition Proportional NorlundB n α z UnitStep n z n z n k 0 n Binomial n k -2 z -1 k α k 1 2 k r 1 k -1 StirlingS1 k 1 r j 1 k -1 j k j r -1 k α -1 Binomial k j Subscript p j k -1 2 k r 1 k -2 StirlingS1 k 1 r j 1 k -1 j k j r -1 k 1 α 2 -1 Binomial k j Subscript p j k Rule 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 n 0 [/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[SuperscriptBox["z", "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], " ", "z"]], ")"]], RowBox[List["-", "k"]]], " ", SuperscriptBox["\[Alpha]", "k"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["2", "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "k"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", "r"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["j", RowBox[List["r", "-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", "k"]]]]], "\[Alpha]"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], RowBox[List[RowBox[List["-", "2"]], "+", "k"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", "r"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["j", RowBox[List["1", "+", "r", "-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", "k"]]]]], SuperscriptBox["\[Alpha]", "2"]]]]]]]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Alpha]", "]"]], "\[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