n | 0 | 1 1 | 1 - x 2 | 1/2 (x^2 - 4 x + 2) 3 | 1/6 (-x^3 + 9 x^2 - 18 x + 6)

n | 0 | 1 | 2 | 3 |

L_n(0) - x L_(n - 1)^1(0) + 1/2 x^2 L_(n - 2)^2(0) - 1/6 x^3 L_(n - 3)^3(0) + 1/24 x^4 L_(n - 4)^4(0) + O(x^5) (Taylor series)

x^n ((-1)^n/Γ(n + 1) - ((-1)^n n^2)/(Γ(n + 1) x) + ((-1)^n (n - 1)^2 n^2)/(2 Γ(n + 1) x^2) - ((-1)^n (n - 2)^2 (n - 1)^2 n^2)/(6 Γ(n + 1) x^3) + ((-1)^n (n - 3)^2 (n - 2)^2 (n - 1)^2 n^2)/(24 Γ(n + 1) x^4) - ((-1)^n (n - 4)^2 (n - 3)^2 (n - 2)^2 (n - 1)^2 n^2)/(120 Γ(n + 1) x^5) + O((1/x)^6)) + e^x x^(-n) (1/(Γ(-n) x) + (n + 1)^2/(Γ(-n) x^2) + ((n + 1)^2 (n + 2)^2)/(2 Γ(-n) x^3) + ((n + 1)^2 (n + 2)^2 (n + 3)^2)/(6 Γ(-n) x^4) + ((n + 1)^2 (n + 2)^2 (n + 3)^2 (n + 4)^2)/(24 Γ(-n) x^5) + ((n + 1)^2 (n + 2)^2 (n + 3)^2 (n + 4)^2 (n + 5)^2)/(120 Γ(-n) x^6) + O((1/x)^7))

d/dx(L_n(x)) = -L_(n - 1)^1(x)

integral L_n(x) dx = L_n(x) - L_(n + 1)(x) + constant

L_n(x) = L_n^0(x)

L_n(x) = lim_(b->∞) P_n^(0, b)(1 - (2 x)/b)

L_n(x) = sum_(k=0)^n c_k L_k^μ(x) for c_k = (-μ)_(-k + n)/((-k + n)!)

L_n(x) = sum_(k=0)^n ((-1)^k x^k binomial(n, k))/(k!)

L_n(x) = sum_(k=0)^n (x^k (-n)_k)/(k!)^2

L_n(x) = sum_(k=0)^n ((-1)^k L_(-k + n)^k(z_0) (x - z_0)^k)/(k!)

L_n(x) = 1/(2 π) integral_0^(2 π) e^(-e^(i t) x) (1 + e^(-i t))^n dt

L_n(x) = e^x/(n!) integral_0^∞ e^(-t) t^n J_0(2 sqrt(t x)) dt

L_n(x) = -i/(2 π)∮_L (e^(-t) ((t + x)/t)^n)/t dt

integral L_n(x) dx = L_n(x) - L_(1 + n)(x)

integral z^(α - 1) L_n(c z) dz = z^α Γ(α) _2 F^~_2(-n, α;1, 1 + α;c z)

integral x^(α - 1) L_n(x) dx = (_2 F_2(-n, α;1, 1 + α;x) x^α)/α