n | 0 | 1 1 | 2 x 2 | 4 x^2 - 2 3 | 8 x^3 - 12 x

n | 0 | 1 | 2 | 3 |

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

x^(n - 1) (i^n 2^(n - 1) (csc((n π)/2) - i sec((n π)/2)) sin(n π) x - (i^n 2^(n - 3) (n - 1) n (csc((n π)/2) - i sec((n π)/2)) sin(n π))/x - (i^(n + 1) 2^(n - 6) (n - 3) (n - 2) (n - 1) n (i csc((n π)/2) + sec((n π)/2)) sin(n π))/x^3 + (i^(n + 1) 2^(n - 8) (n - 5) (n - 4) (n - 3) (n - 2) (n - 1) n (i csc((n π)/2) + sec((n π)/2)) sin(n π))/(3 x^5) + O((1/x)^6))

d/dx(H_n(x)) = 2 n H_(n - 1)(x)

integral H_n(x) dx = (H_(n + 1)(x))/(2 n + 2) + constant

H_n(x) = D_n(x sqrt(2)) 2^(n/2) e^(x^2/2)

H_n(x) = n/2 ! L_(n/2)^(-1/2)(x^2) (-1)^(n/2) 2^n

H_n(x) = n! (lim_(a->∞) (P_n^(a, a)(x/sqrt(a)))/a^(n/2)) 2^n

H_n(x) = sum_(k=0)^n 2^k binomial(n, k) H_(-k + n)(z_0) (x - z_0)^k

H_n(x) = n! sum_(k=0)^floor(n/2) ((-1)^k 2^(-2 k + n) x^(-2 k + n))/(k! (-2 k + n)!)

H_n(x) = 2^n x^n sum_(k=0)^floor(n/2) ((-1)^k x^(-2 k) ((1 - n)/2)_k (-n/2)_k)/(k!)

H_n(x) = 2^n/sqrt(π) integral_(-∞)^∞ e^(-t^2) (i t + x)^n dt

H_n(x) = (2^n e^((3 i n π)/2))/sqrt(π) integral_(-∞)^∞ e^(-(t - i x)^2) t^n dt

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

H_n(x) = -(i n!)/(2 π)∮_L e^(-t (t - 2 x)) t^(-1 - n) dt

integral H_n(x) dx = (H_(1 + n)(x))/(2 (1 + n))

integral H_n(x) dx = n! sum_(k=0)^floor(n/2) ((-1)^k (2^(n - 2 k) x^(1 - 2 k + n)))/(k! (n - 2 k + 1)!)

integral x^(α - 1) H_n(x) dx = n! sum_(k=0)^floor(n/2) ((-1)^k (2^(n - 2 k) x^(n - 2 k + α)))/((n - 2 k + α) k! (n - 2 k)!)