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 Gamma

 http://functions.wolfram.com/06.05.06.0031.01

 Input Form

 Gamma[-n + \[Epsilon]] \[Proportional] ((-1)^n/(n! \[Epsilon])) (1 + Pi \[Epsilon] Sum[(j + 1) Sum[(((-1)^(r + j) Binomial[j, r])/(1 + r)) Subscript[p, r, j] Subscript[c, k - j] \[Epsilon]^k, {r, 0, j}], {k, 0, Infinity}, {j, 0, k + 1}]) /; Element[n, Integers] && n >= 0 && (\[Epsilon] -> 0) && Subscript[c, 2 k] == 0 && Subscript[c, 2 k + 1] == ((-1)^k 2 (2^(2 k + 1) - 1) BernoulliB[2 k + 2] Pi^(2 k + 1))/(2 k + 2)! && Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/k) Sum[(j m - k + m) Subscript[b, m] Subscript[p, j, k - m], {m, 1, k}] && Subscript[b, k] == Derivative[k][Gamma][n + 1]/(n! k!)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], "+", "\[Epsilon]"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List[RowBox[List["n", "!"]], "\[Epsilon]"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Pi]", " ", "\[Epsilon]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "+", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "j"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "+", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "r"]], "]"]]]], RowBox[List["1", "+", "r"]]], SubscriptBox["p", RowBox[List["r", ",", "j"]]], SubscriptBox["c", RowBox[List["k", "-", "j"]]], " ", SuperscriptBox["\[Epsilon]", "k"]]]]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["(", RowBox[List["\[Epsilon]", "\[Rule]", "0"]], ")"]], "\[And]", RowBox[List[SubscriptBox["c", RowBox[List["2", "k"]]], "\[Equal]", "0"]], "\[And]", RowBox[List[SubscriptBox["c", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], "-", "1"]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["2", "k"]], "+", "2"]], "]"]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "2"]], ")"]], "!"]]]]], "\[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["b", "m"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]]]]]], "\[And]", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["n", "+", "1"]], "]"]], RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["k", "!"]]]]]]]]]]]]]

 MathML Form

 Γ ( ϵ - n ) ( - 1 ) n n ! ϵ ( 1 + π ϵ k = 0 j = 0 k + 1 ( j + 1 ) r = 0 j ( - 1 ) j + r ( j r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["r", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] r + 1 p r , j c k - j ϵ k ) /; n ( ϵ "\[Rule]" 0 ) c 2 k 0 c 2 k + 1 ( - 1 ) k 2 ( 2 2 k + 1 - 1 ) B TagBox["B", BernoulliB] 2 k + 2 π 2 k + 1 ( 2 k + 2 ) ! p j , 0 1 p j , k 1 k m = 1 k ( j m + m - k ) b m p j , k - m b k Γ ( k ) TagBox[RowBox[List["(", "k", ")"]], Derivative] ( n + 1 ) n ! k ! Condition Proportional Gamma ϵ -1 n -1 n n ϵ -1 1 ϵ j 0 k 1 k 0 j 1 r 0 j -1 j r Binomial j r r 1 -1 Subscript p r j Subscript c k -1 j ϵ k n Rule ϵ 0 Subscript c 2 k 0 Subscript c 2 k 1 -1 k 2 2 2 k 1 -1 BernoulliB 2 k 2 2 k 1 2 k 2 -1 Subscript p j 0 1 Subscript p j k 1 k -1 m 1 k j m m -1 k Subscript b m Subscript p j k -1 m Subscript b k D Gamma n 1 n 1 k n k -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n_"]], "+", "\[Epsilon]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[Pi]", " ", "\[Epsilon]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "+", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "j"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "+", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "r"]], "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["r", ",", "j"]]], " ", SubscriptBox["c", RowBox[List["k", "-", "j"]]], " ", SuperscriptBox["\[Epsilon]", "k"]]], RowBox[List["1", "+", "r"]]]]]]]]]]]]]]], ")"]]]], RowBox[List[RowBox[List["n", "!"]], " ", "\[Epsilon]"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["(", RowBox[List["\[Epsilon]", "\[Rule]", "0"]], ")"]], "&&", RowBox[List[SubscriptBox["c", RowBox[List["2", " ", "k"]]], "\[Equal]", "0"]], "&&", RowBox[List[SubscriptBox["c", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "2", " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], "-", "1"]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", "2"]], "]"]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "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["b", "m"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]], "k"]]], "&&", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["n", "+", "1"]], "]"]], RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["k", "!"]]]]]]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02