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 NorlundB

 http://functions.wolfram.com/05.17.20.0001.01

 Input Form

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

 B TagBox["B", NorlundB] n ( α ) ( z ) α n ! k = 0 n c n - k z k k ! ( n - k ) ! /; c k ( α ) k + 1 TagBox[SubscriptBox[RowBox[List["(", "\[Alpha]", ")"]], RowBox[List["k", "+", "1"]]], Pochhammer] j = 0 k ( - 1 ) j ( j + 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]] ( - ( j + α ) ψ TagBox["\[Psi]", PolyGamma] ( α ) + ( j + α ) ψ TagBox["\[Psi]", PolyGamma] ( k + α + 1 ) - 1 ) 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 NorlundB n α z n k 0 n Subscript c n -1 k z k k n -1 k -1 Subscript c k Pochhammer α k 1 j 0 k -1 j j α 2 -1 Binomial k j -1 j α PolyGamma α j α PolyGamma k α 1 -1 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["\[Alpha]_"]]], 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["n", "-", "k"]]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "k"], "\[Equal]", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Alpha]", ",", RowBox[List["1", "+", "k"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List[RowBox[List["(", RowBox[List["j", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["PolyGamma", "[", "\[Alpha]", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["j", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", "\[Alpha]"]], "]"]]]]]], ")"]], " ", SubscriptBox["p", RowBox[List["j", ",", "k"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["j", "+", "\[Alpha]"]], ")"]], "2"]]]]]]]], "&&", 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