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 NorlundB

 http://functions.wolfram.com/05.17.20.0004.01

 Input Form

 D[NorlundB[n, \[Alpha], z], {\[Alpha], m}] == n! Sum[(Subscript[c, n - k, m]/(k! (n - k)!)) z^k, {k, 0, n}] /; Subscript[c, k, m] == Sum[(-1)^j Binomial[k, j] Sum[Binomial[m, r] (m - r)! \[Alpha]^(1 - r) (\[Alpha] + j)^(-1 - m + r) Sum[(-1)^(i + k + m - r) StirlingS1[k + 1, i + 1] Pochhammer[i - r + 2, r] \[Alpha]^i Subscript[p, j, k], {i, 0, k}], {r, 0, m}], {j, 0, k}] && 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["\[Alpha]", ",", "m"]], "}"]]], RowBox[List["NorlundB", "[", RowBox[List["n", ",", "\[Alpha]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["n", "!"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[SubscriptBox["c", RowBox[List[RowBox[List["n", "-", "k"]], ",", "m"]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]]], SuperscriptBox["z", "k"]]]]]]]]], "/;", "\[IndentingNewLine]", RowBox[List[RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "m"]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "r"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "r"]], ")"]], "!"]], " ", SuperscriptBox["\[Alpha]", RowBox[List["1", "-", "r"]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[Alpha]", "+", "j"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "m", "+", "r"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "+", "k", "+", "m", "-", "r"]]], " ", RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["k", "+", "1"]], ",", RowBox[List["i", "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["i", "-", "r", "+", "2"]], ",", "r"]], "]"]], " ", SuperscriptBox["\[Alpha]", "i"], SubscriptBox["p", RowBox[List["j", ",", "k"]]]]]]]]]]]]]]]]], "\[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"]]]]]]]]

 MathML Form

 m B TagBox["B", NorlundB] n ( α ) ( z ) α m n ! k = 0 n c n - k , m z k k ! ( n - k ) ! /; c k , m j = 0 k ( - 1 ) j ( 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]] r = 0 m ( m r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["r", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( m - r ) ! α 1 - r ( j + α ) r - m - 1 i = 0 k ( - 1 ) i + k + m - r S TagBox["S", StirlingS1] k + 1 ( i + 1 ) ( i - r + 2 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["i", "-", "r", "+", "2"]], ")"]], "r"], Pochhammer] α i p j , k 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 α m NorlundB n α z n k 0 n Subscript c n -1 k m z k k n -1 k -1 Subscript c k m j 0 k -1 j Binomial k j r 0 m Binomial m r m -1 r α 1 -1 r j α r -1 m -1 i 0 k -1 i k m -1 r StirlingS1 k 1 i 1 Pochhammer i -1 r 2 r α i Subscript p j k 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["\[Alpha]_", ",", "m_"]], "}"]]]]], RowBox[List["NorlundB", "[", RowBox[List["n_", ",", "\[Alpha]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SubscriptBox["c", RowBox[List[RowBox[List["n", "-", "k"]], ",", "m"]]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", RowBox[List["k", ",", "m"]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "r"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "r"]], ")"]], "!"]], " ", SuperscriptBox["\[Alpha]", RowBox[List["1", "-", "r"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Alpha]", "+", "j"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "m", "+", "r"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["i", "+", "k", "+", "m", "-", "r"]]], " ", RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["k", "+", "1"]], ",", RowBox[List["i", "+", "1"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["i", "-", "r", "+", "2"]], ",", "r"]], "]"]], " ", SuperscriptBox["\[Alpha]", "i"], " ", SubscriptBox["p", RowBox[List["j", ",", "k"]]]]]]]]]]]]]]]]], "&&", 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