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 LegendreP

 http://functions.wolfram.com/07.09.20.0011.02

 Input Form

 D[LegendreP[\[Nu], \[Mu], 3, z], {\[Nu], m}] == ((z + 1)^(\[Mu]/2)/(z - 1)^(\[Mu]/2)) Sum[(1/(Gamma[1 - \[Mu] + k] k!)) ((1 - z)/2)^k Sum[Binomial[m, j] Sum[StirlingS1[k, i] Pochhammer[i - j + 1, j] \[Nu]^(i - j) Sum[(-1)^r StirlingS1[k, r] Pochhammer[r - m + j + 1, m - j] (\[Nu] + 1)^(r - m + j), {r, 1, k}], {i, 1, k}], {j, 0, m}], {k, 0, Infinity}] /; Abs[(1 - z)/2] < 1 && Element[m, Integers] && m >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Nu]", ",", "m"]], "}"]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "j"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["i", "-", "j", "+", "1"]], ",", "j"]], "]"]], " ", SuperscriptBox["\[Nu]", RowBox[List["i", "-", "j"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "r"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["r", "-", "m", "+", "j", "+", "1"]], ",", RowBox[List["m", "-", "j"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], RowBox[List["r", "-", "m", "+", "j"]]]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 m 𝔓 TagBox["\[GothicCapitalP]", LegendreQ] ν μ ( z ) ν m ( z + 1 ) μ / 2 ( z - 1 ) μ / 2 k = 0 1 Γ ( k - μ + 1 ) k ! ( 1 - z 2 ) k j = 0 m ( m j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] i = 1 k S TagBox["S", StirlingS1] k ( i ) ( i - j + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["i", "-", "j", "+", "1"]], ")"]], "j"], Pochhammer] ν i - j r = 1 k ( - 1 ) r S TagBox["S", StirlingS1] k ( r ) ( j - m + r + 1 ) m - j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "m", "+", "r", "+", "1"]], ")"]], RowBox[List["m", "-", "j"]]], Pochhammer] ( ν + 1 ) j - m + r /; "\[LeftBracketingBar]" 1 - z 2 "\[RightBracketingBar]" < 1 m Condition ν m Subscript LegendreQ 𝔓 ν μ z z 1 μ 2 -1 z -1 μ 2 -1 -1 k 0 1 Gamma k -1 μ 1 k -1 1 -1 z 2 -1 k j 0 m Binomial m j i 1 k StirlingS1 k i Pochhammer i -1 j 1 j ν i -1 j r 1 k -1 r StirlingS1 k r Pochhammer j -1 m r 1 m -1 j ν 1 j -1 m r 1 -1 z 2 -1 1 m [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Nu]_", ",", "m_"]], "}"]]]]], RowBox[List["LegendreP", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "z"]], "2"], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "m"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "j"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["i", "-", "j", "+", "1"]], ",", "j"]], "]"]], " ", SuperscriptBox["\[Nu]", RowBox[List["i", "-", "j"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["StirlingS1", "[", RowBox[List["k", ",", "r"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["r", "-", "m", "+", "j", "+", "1"]], ",", RowBox[List["m", "-", "j"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], RowBox[List["r", "-", "m", "+", "j"]]]]]]]]]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Mu]", "+", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["\[Mu]", "/", "2"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[RowBox[List["1", "-", "z"]], "2"], "]"]], "<", "1"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29