n | 0 | 1 1 | x 2 | 1/2 (3 x^2 - 1) 3 | 1/2 (5 x^3 - 3 x)

n | 0 | 2 | 3 |

sqrt(π)/(Γ(1/2 - n/2) Γ(n/2 + 1)) + (sqrt(π) n (n + 1) x)/(2 Γ(1 - n/2) Γ((n + 3)/2)) + (sqrt(π) (n - 1) n (n + 1) (n + 2) x^2)/(8 Γ(3/2 - n/2) Γ(n/2 + 2)) + (sqrt(π) (n - 2) (n - 1) n (n + 1) (n + 2) (n + 3) x^3)/(48 Γ(2 - n/2) Γ((n + 5)/2)) + (sqrt(π) (n - 3) (n - 2) (n - 1) n (n + 1) (n + 2) (n + 3) (n + 4) x^4)/(384 Γ(5/2 - n/2) Γ(n/2 + 3)) + O(x^5) (Taylor series)

(x - 1)^n ((2^(-n) Γ(2 n + 1))/Γ(n + 1)^2 + (2^(-n) n Γ(2 n + 1))/(Γ(n + 1)^2 x) + (2^(-n) n^3 Γ(2 n + 1))/((2 n - 1) Γ(n + 1)^2 x^2) + (2^(-n) n (n^3 + n^2 + 2 n - 1) Γ(2 n + 1))/(3 (2 n - 1) Γ(n + 1)^2 x^3) + (2^(-n - 1) n^3 (n^3 + n^2 + 5 n - 13) Γ(2 n + 1))/(3 (4 n^2 - 8 n + 3) Γ(n + 1)^2 x^4) + (2^(-n - 1) n (n^6 + 3 n^5 + 17 n^4 - 15 n^3 - 6 n^2 - 48 n + 18) Γ(2 n + 1))/(15 (4 n^2 - 8 n + 3) Γ(n + 1)^2 x^5) + O((1/x)^6)) + (x - 1)^(-n - 1) ((2^(n + 1) Γ(-2 n - 1))/Γ(-n)^2 - (2^(n + 1) (n + 1) Γ(-2 n - 1))/(Γ(-n)^2 x) + (2^(n + 1) (n + 1)^3 Γ(-2 n - 1))/((2 n + 3) Γ(-n)^2 x^2) - (2^(n + 1) (n + 1) (n^3 + 2 n^2 + 3 n + 3) Γ(-2 n - 1))/(3 ((2 n + 3) Γ(-n)^2) x^3) + (2^n (n + 1)^3 (n^3 + 2 n^2 + 6 n + 18) Γ(-2 n - 1))/(3 (2 n + 3) (2 n + 5) Γ(-n)^2 x^4) - (2^n (n^7 + 4 n^6 + 20 n^5 + 90 n^4 + 199 n^3 + 266 n^2 + 230 n + 90) Γ(-2 n - 1))/(15 ((2 n + 3) (2 n + 5) Γ(-n)^2) x^5) + O((1/x)^6))

d/dx(P_n(x)) = -((n + 1) (x P_n(x) - P_(n + 1)(x)))/(x^2 - 1)

integral P_n(x) dx = (P_(n + 1)(x) - P_(n - 1)(x))/(2 n + 1) + constant

P_n(x) = P_n^0(x)

P_n(x) = P_n^(0, 0)(x)

P_n(x) = P_n^0(x)

P_n(x) = 2^(-n) (-1 + x)^n sum_(k=0)^n ((1 + x)/(-1 + x))^k binomial(n, k)^2

P_n(x) = 2^(-n) sum_(k=0)^floor(n/2) (-1)^k x^(-2 k + n) binomial(n, k) binomial(-2 k + 2 n, n)

P_n(x) = (-1)^n sum_(k=0)^n (2^(-k) (1 + x)^k (-n)_k (1 + n)_k)/(k!)^2

P_n(x) = 2^n/π integral_(-∞)^∞ (1 + t^2)^(-1 - n) (i t + x)^n dt

P_n(x) = 1/π integral_0^π (x - sqrt(-1 + x^2) cos(t))^n dt

P_n(x) = 1/π integral_0^π (x + i sqrt(1 - x^2) cos(t))^n dt

integral P_n(x) dx = (P_(n + 1)(x) - P_(n - 1)(x))/(1 + 2 n)

integral(1 - x^2)^(1/2 (-3 - n)) P_n(x) dx = ((1 - x^2)^(1/2 (-1 - n)) P_(1 + n)(x))/(1 + n)

integral(1 - x^2)^(-1 + n/2) P_n(x) dx = -((1 - x^2)^(n/2) P_(-1 + n)(x))/n