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 PolyLog

 http://functions.wolfram.com/10.09.17.0002.01

 Input Form

 PolyLog[n, p, z] == (-1)^n Sum[(-1)^k Sum[(Log[-z]^j Binomial[n + k - j - 1, k - j] PolyLog[n + k - j, p - k, 1/z])/j!, {j, 0, k}], {k, 0, p - 1}] + (-1)^p (Sum[(Log[-z]^j c[n - j, p])/j!, {j, 0, n - 1}] + Log[-z]^(n + p)/(n + p)!) /; c[n, p] == (1 - (-1)^n) (-1)^p PolyLog[n, p, -1] - (-1)^n Sum[Binomial[n + j - 1, j] (-1)^(p - j) PolyLog[n + j, p - j, -1], {j, 1, p - 1}] && Element[n, Integers] && n > 0 && Element[p, Integers] && p > 0 && !IntervalMemberQ[Interval[{0, 1}], z]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyLog", "[", RowBox[List["n", ",", "p", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k", "-", "j", "-", "1"]], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["n", "+", "k", "-", "j"]], ",", RowBox[List["p", "-", "k"]], ",", FractionBox["1", "z"]]], "]"]]]], RowBox[List["j", "!"]]]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "j"], " ", RowBox[List["c", "[", RowBox[List[RowBox[List["n", "-", "j"]], ",", "p"]], "]"]]]], RowBox[List["j", "!"]]]]], "+", FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], RowBox[List["n", "+", "p"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "p"]], ")"]], "!"]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["c", "[", RowBox[List["n", ",", "p"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["PolyLog", "[", RowBox[List["n", ",", "p", ",", RowBox[List["-", "1"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "j", "-", "1"]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "j"]]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["n", "+", "j"]], ",", RowBox[List["p", "-", "j"]], ",", RowBox[List["-", "1"]]]], "]"]]]]]]]]]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", ">", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]]]]

 MathML Form

 S PolyLog n , p ( z ) ( - 1 ) p ( log n + p ( - z ) ( n + p ) ! + j = 0 n - 1 log j ( - z ) c ( n - j , p ) j ! ) + ( - 1 ) n k = 0 p - 1 ( - 1 ) k j = 0 k log j ( - z ) j ! ( n - j + k - 1 k - j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "j", "+", "k", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["k", "-", "j"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] S TagBox["S", StirlingS1] n + k - j p - k ( 1 z ) /; c ( n , p ) ( 1 - ( - 1 ) n ) ( - 1 ) p S TagBox["S", StirlingS1] n p ( - 1 ) - ( - 1 ) n j = 1 p - 1 ( j + n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "+", "n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - 1 ) p - j S TagBox["S", StirlingS1] n + j p - j ( - 1 ) n + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] p + z ( 0 , 1 ) S PolyLog n , p ( z ) ( - 1 ) p ( log n + p ( - z ) ( n + p ) ! + j = 0 n - 1 log j ( - z ) c ( n - j , p ) j ! ) + ( - 1 ) n k = 0 p - 1 ( - 1 ) k j = 0 k log j ( - z ) j ! ( n - j + k - 1 k - j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "j", "+", "k", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["k", "-", "j"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] S TagBox["S", StirlingS1] n + k - j p - k ( 1 z ) /; c ( n , p ) ( 1 - ( - 1 ) n ) ( - 1 ) p S TagBox["S", StirlingS1] n p ( - 1 ) - ( - 1 ) n j = 1 p - 1 ( j + n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "+", "n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - 1 ) p - j S TagBox["S", StirlingS1] n + j p - j ( - 1 ) n + TagBox[SuperscriptBox["\[DoubleStruckCapitalN]", "+"], Function[Integers]] p + z ( 0 , 1 ) [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyLog", "[", RowBox[List["n_", ",", "p_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["p", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k", "-", "j", "-", "1"]], ",", RowBox[List["k", "-", "j"]]]], "]"]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["n", "+", "k", "-", "j"]], ",", RowBox[List["p", "-", "k"]], ",", FractionBox["1", "z"]]], "]"]]]], RowBox[List["j", "!"]]]]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "j"], " ", RowBox[List["c", "[", RowBox[List[RowBox[List["n", "-", "j"]], ",", "p"]], "]"]]]], RowBox[List["j", "!"]]]]], "+", FractionBox[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], RowBox[List["n", "+", "p"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "p"]], ")"]], "!"]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["c", "[", RowBox[List["n", ",", "p"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["PolyLog", "[", RowBox[List["n", ",", "p", ",", RowBox[List["-", "1"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "j", "-", "1"]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "-", "j"]]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["n", "+", "j"]], ",", RowBox[List["p", "-", "j"]], ",", RowBox[List["-", "1"]]]], "]"]]]]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", ">", "0"]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["0", ",", "1"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]]]

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

 2001-10-29