P_n^(a, b)(0) + 1/2 x (a + b + n + 1) P_(n - 1)^(a + 1, b + 1)(0) + 1/8 x^2 (a + b + n + 1) (a + b + n + 2) P_(n - 2)^(a + 2, b + 2)(0) + 1/48 x^3 (a + b + n + 1) (a + b + n + 2) (a + b + n + 3) P_(n - 3)^(a + 3, b + 3)(0) + 1/384 x^4 (a + b + n + 1) (a + b + n + 2) (a + b + n + 3) (a + b + n + 4) P_(n - 4)^(a + 4, b + 4)(0) + O(x^5) (Taylor series)

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

integral P_n^(a, b)(x) dx = (2 P_(n + 1)^(a - 1, b - 1)(x))/(a + b + n) + constant

P_n^(a, b)(x) = sum_(k=0)^n c_k P_k(x) for c_k = 1/2 + k integral_(-1)^1 P_n^(a, b)(t) P_k(t) dt

P_n^(a, b)(x) = sum_(k=0)^n c_k L_k(x) for c_k = integral_0^∞ e^(-t) P_n^(a, b)(t) L_k(t) dt

P_n^(a, b)(x) = sum_(k=0)^n c_k U_k(x) for 2 integral_(-1)^1 sqrt(1 - t^2) U_k(t) P_n^(a, b)(t) dt = π c_k

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

P_n^(a, b)(x) = sum_(k=0)^n (2^(-k) P_(-k + n)^(a + k, b + k)(z_0) (1 + a + b + n)_k (x - z_0)^k)/(k!)

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

P_n^(a, b)(x) = (2^(-n) Γ(1 + a + n))/(Γ(-b - n) Γ(1 + n) Γ(1 + a + b + n)) integral_0^1 (1 - t)^(-1 - b - n) t^(a + b + n) (2 + t (-1 + x))^n dt for (Re(b + n)<0 and 1 + Re(a + b + n)>0 and abs(arg(1 + x))<π)

P_n^(a, b)(x) = -(i 2^(-1 - n) (1 - x)^(-a) (1 + x)^(-b))/π∮_L ((1 - t)^a (1 + t)^b ((-1 + t^2)/(t - x))^n)/(t - x) dt

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

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

integral(1 - x)^a (1 + x)^b P_n^(a, b)(x) dx = (P_(n - 1)^(a + 1, b + 1)(x) (-1) ((1 - x)^(a + 1) (1 + x)^(b + 1)))/(2 n) for n>0