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 LaguerreL

 http://functions.wolfram.com/05.02.06.0021.01

 Input Form

 LaguerreL[n, z] \[Proportional] (E^(z/2)/(z^(1/4) n^(1/4) Sqrt[Pi])) (Cos[Pi/4 - 2 Sqrt[(n + 1/2) z]] + Sum[(Sum[(((-1)^(j + r + s) 2^(2 j - 2 k + s) z^(-j + k - 2 r - s))/ (s! Pochhammer[1/2, r])) Subscript[A, 2 (k - j - r - s)] Cos[Pi (-(1/4) + j - k + r + s) + 2 Sqrt[(n + 1/2) z]] BernoulliB[j] KroneckerDelta[j] Pochhammer[1/4 - j + k - s, s] Pochhammer[1/4 - j + k - r - s, r] Pochhammer[3/4 - j + k - r - s, r] Pochhammer[1/4 + j - k + r + s, r] Pochhammer[ 3/4 + j - k + r + s, r], {j, 0, k}, {r, 0, k - j}, {s, 0, k - j - r}] - (1/z) 2 Sum[(((-1)^(j + r + s) 2^(2 j - 2 k + s) z^(k - j - 2 r - s))/ (s! Pochhammer[3/2, r])) Subscript[A, 2 (k - j - r - s) - 1] BernoulliB[j] KroneckerDelta[j] Pochhammer[1/4 - j + k - s, s] Pochhammer[1/4 - j + k - r - s, r] Pochhammer[3/4 - j + k - r - s, r] Pochhammer[1/4 + j - k + r + s, 1 + r] Pochhammer[ 3/4 + j - k + r + s, 1 + r] Sin[Pi (1/4 + j - k + r + s) + 2 Sqrt[(n + 1/2) z]], {j, 0, k - 1}, {r, 0, k - j - 1}, {s, 0, k - j - r - 1}])/n^k, {k, 1, Infinity}] + (Sqrt[z]/(2 Sqrt[n])) Sum[(((-1)^(j + r + s) 2^(2 j - 2 k + s) z^(k - j - 2 r - s))/(n^k (s! Pochhammer[3/2, r]))) BernoulliB[j] KroneckerDelta[j] Pochhammer[3/4 - j + k - s, s] Pochhammer[3/4 - j + k - r - s, r] Pochhammer[5/4 - j + k - r - s, r] Pochhammer[-(1/4) + j - k + r + s, r] Pochhammer[1/4 + j - k + r + s, r] ((1 + 2 r) Subscript[A, 2 (k - j - r - s) + 1] Cos[Pi (-(3/4) + j - k + r + s) + 2 Sqrt[(n + 1/2) z]] - (((-1 + 4 j - 4 k + 8 r + 4 s) (1 + 4 j - 4 k + 8 r + 4 s))/(8 z)) Subscript[A, 2 (k - j - r - s)] Sin[Pi (-(1/4) + j - k + r + s) + 2 Sqrt[(n + 1/2) z]]), {k, 0, Infinity}, {j, 0, k}, {r, 0, k - j}, {s, 0, k - j - r}]) /; (n -> Infinity) && Subscript[A, 0] == 1 && Subscript[A, 1] == 0 && Subscript[A, 2] == 1/2 && Subscript[A, m] == ((m - 1)/m) Subscript[A, m - 2] - (2 n + 1) Subscript[A, m - 3] && Element[m, Integers] && m > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", "/", "2"]]], " ", SuperscriptBox["z", RowBox[List["-", FractionBox["1", "4"]]]], SuperscriptBox["n", RowBox[List["-", FractionBox["1", "4"]]]]]], SqrtBox["\[Pi]"]], RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox["\[Pi]", "4"], "-", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", FractionBox["1", "2"]]], ")"]], " ", "z"]]]]]]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["n", RowBox[List["-", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], RowBox[List["k", "-", "j"]]], 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RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "r"]]]], ")"]], " ", SubscriptBox["A", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "-", "j", "-", "r", "-", "s"]], ")"]]]], "+", "1"]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], "+", "j", "-", "k", "+", "r", "+", "s"]], ")"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["n", "+", FractionBox["1", "2"]]], ")"]], " ", "z"]]]]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "j"]], "-", RowBox[List["4", " ", "k"]], "+", RowBox[List["8", " ", "r"]], "+", RowBox[List["4", " ", "s"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "j"]], "-", RowBox[List["4", " ", "k"]], "+", RowBox[List["8", " ", "r"]], "+", RowBox[List["4", " 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RowBox[List[RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], "m"], SubscriptBox["A", RowBox[List["m", "-", "2"]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "1"]], ")"]], SubscriptBox["A", RowBox[List["m", "-", "3"]]]]]]]]], "\[And]", RowBox[List["Element", "[", RowBox[List["m", ",", "Integers"]], "]"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 L n ( z ) z / 2 z - 1 4 n - 1 4 π ( cos ( π 4 - 2 ( n + 1 2 ) z ) + k = 1 n - k ( j = 0 k r = 0 k - j s = 0 k - j - r ( - 1 ) j + r + s 2 2 j - 2 k + s z k - j - 2 r - s s ! ( 1 2 ) r TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "r"], Pochhammer] A 2 ( k - j - r - s ) cos ( π ( j - k + r + s - 1 4 ) + 2 ( n + 1 2 ) z ) B TagBox["B", BernoulliB] j δ KroneckerDelta j ( k - j - s + 1 4 ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "j", "-", "s", "+", FractionBox["1", "4"]]], ")"]], "s"], Pochhammer] ( k - j - r - s + 1 4 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "j", "-", "r", "-", "s", "+", FractionBox["1", "4"]]], ")"]], "r"], Pochhammer] ( k - j - r - s + 3 4 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "j", "-", "r", "-", "s", "+", FractionBox["3", "4"]]], ")"]], "r"], Pochhammer] ( j - k + r + s + 1 4 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "k", "+", "r", "+", "s", "+", FractionBox["1", "4"]]], ")"]], "r"], Pochhammer] ( j - k + r + s + 3 4 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "k", "+", "r", "+", "s", "+", FractionBox["3", "4"]]], ")"]], "r"], Pochhammer] - 2 z j = 0 k - 1 r = 0 k - j - 1 s = 0 k - j - r - 1 ( - 1 ) j + r + s 2 2 j - 2 k + s z k - j - 2 r - s s ! ( 3 2 ) r TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "r"], Pochhammer] A 2 ( k - j - r - s ) - 1 B TagBox["B", BernoulliB] j δ KroneckerDelta j ( k - j - s + 1 4 ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "j", "-", "s", "+", FractionBox["1", "4"]]], ")"]], "s"], Pochhammer] ( k - j - r - s + 1 4 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "j", "-", "r", "-", "s", "+", FractionBox["1", "4"]]], ")"]], "r"], Pochhammer] ( k - j - r - s + 3 4 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "j", "-", "r", "-", "s", "+", FractionBox["3", "4"]]], ")"]], "r"], Pochhammer] ( j - k + r + s + 1 4 ) r + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "k", "+", "r", "+", "s", "+", FractionBox["1", "4"]]], ")"]], RowBox[List["r", "+", "1"]]], Pochhammer] ( j - k + r + s + 3 4 ) r + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "k", "+", "r", "+", "s", "+", FractionBox["3", "4"]]], ")"]], RowBox[List["r", "+", "1"]]], Pochhammer] sin ( π ( j - k + r + s + 1 4 ) + 2 ( n + 1 2 ) z ) ) + z 2 n k = 0 j = 0 k r = 0 k - j s = 0 k - j - r ( - 1 ) j + r + s 2 2 j - 2 k + s n - k z k - j - 2 r - s s ! ( 3 2 ) r TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "r"], Pochhammer] B TagBox["B", BernoulliB] j δ KroneckerDelta j ( k - j - s + 3 4 ) s TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "j", "-", "s", "+", FractionBox["3", "4"]]], ")"]], "s"], Pochhammer] ( k - j - r - s + 3 4 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "j", "-", "r", "-", "s", "+", FractionBox["3", "4"]]], ")"]], "r"], Pochhammer] ( k - j - r - s + 5 4 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "j", "-", "r", "-", "s", "+", FractionBox["5", "4"]]], ")"]], "r"], Pochhammer] ( j - k + r + s - 1 4 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "k", "+", "r", "+", "s", "-", FractionBox["1", "4"]]], ")"]], "r"], Pochhammer] ( j - k + r + s + 1 4 ) r TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "k", "+", "r", "+", "s", "+", FractionBox["1", "4"]]], ")"]], "r"], Pochhammer] ( ( 2 r + 1 ) A 2 ( k - j - r - s ) + 1 cos ( π ( j - k + r + s - 3 4 ) + 2 ( n + 1 2 ) z ) - ( 4 j - 4 k + 8 r + 4 s - 1 ) ( 4 j - 4 k + 8 r + 4 s + 1 ) 8 z A 2 ( k - j - r - s ) sin ( π ( j - k + r + s - 1 4 ) + 2 ( n + 1 2 ) z ) ) ) /; ( n "\[Rule]" ) A 0 1 A 1 0 A 2 1 2 A m m - 1 m A m - 2 - ( 2 n + 1 ) A m - 3 m + Condition Proportional LaguerreL n z z 2 -1 z -1 1 4 n -1 1 4 1 2 -1 4 -1 -1 2 n 1 2 z 1 2 k 1 n -1 k s 0 k -1 j -1 r r 0 k -1 j j 0 k -1 j r s 2 2 j -1 2 k s z k -1 j -1 2 r -1 s s Pochhammer 1 2 r -1 Subscript A 2 k -1 j -1 r -1 s j -1 k r s -1 1 4 2 n 1 2 z 1 2 BernoulliB j KroneckerDelta j Pochhammer k -1 j -1 s 1 4 s Pochhammer k -1 j -1 r -1 s 1 4 r Pochhammer k -1 j -1 r -1 s 3 4 r Pochhammer j -1 k r s 1 4 r Pochhammer j -1 k r s 3 4 r -1 2 z -1 s 0 k -1 j -1 r -1 r 0 k -1 j -1 j 0 k -1 -1 j r s 2 2 j -1 2 k s z k -1 j -1 2 r -1 s s Pochhammer 3 2 r -1 Subscript A 2 k -1 j -1 r -1 s -1 BernoulliB j KroneckerDelta j Pochhammer k -1 j -1 s 1 4 s Pochhammer k -1 j -1 r -1 s 1 4 r Pochhammer k -1 j -1 r -1 s 3 4 r Pochhammer j -1 k r s 1 4 r 1 Pochhammer j -1 k r s 3 4 r 1 j -1 k r s 1 4 2 n 1 2 z 1 2 z 1 2 2 n 1 2 -1 s 0 k -1 j -1 r r 0 k -1 j j 0 k k 0 -1 j r s 2 2 j -1 2 k s n -1 k z k -1 j -1 2 r -1 s s Pochhammer 3 2 r -1 BernoulliB j KroneckerDelta j Pochhammer k -1 j -1 s 3 4 s Pochhammer k -1 j -1 r -1 s 3 4 r Pochhammer k -1 j -1 r -1 s 5 4 r Pochhammer j -1 k r s -1 1 4 r Pochhammer j -1 k r s 1 4 r 2 r 1 Subscript A 2 k -1 j -1 r -1 s 1 j -1 k r s -1 3 4 2 n 1 2 z 1 2 -1 4 j -1 4 k 8 r 4 s -1 4 j -1 4 k 8 r 4 s 1 8 z -1 Subscript A 2 k -1 j -1 r -1 s j -1 k r s -1 1 4 2 n 1 2 z 1 2 Rule n Subscript A 0 1 Subscript A 1 0 Subscript A 2 1 2 Subscript A m m -1 m -1 Subscript A m -2 -1 2 n 1 Subscript A m -3 m SuperPlus [/itex]

 Rule Form

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

 2007-05-02