Hermite Polynomials
📂Functions Hermite Polynomials Description Hermite Polynomials are defined in several ways as follows.
As solutions to a differential equation Hermite polynomials are defined as the solutions to the following Hermite Differential Equation .
y ′ ′ − 2 x y ′ + 2 n y = 0 , n = 0 , 1 , 2 , ⋯
y^{\prime \prime} -2xy^{\prime} +2ny=0,\quad n=0,1,2,\cdots
y ′′ − 2 x y ′ + 2 n y = 0 , n = 0 , 1 , 2 , ⋯
The following function H n H_{n} H n
H n ( x ) = ( − 1 ) n e x 2 d n d x n e − x 2
H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{ d ^{n}}{ dx^{n} }e^{-x^{2}}
H n ( x ) = ( − 1 ) n e x 2 d x n d n e − x 2
This is known as the Rodrigues’ formula . Meanwhile, the above function is referred to as the physicist’s Hermite polynomials , and the form below is called the probabilist’s Hermite polynomials .
H e n : = ( − 1 ) n e x 2 2 d n d x n e − x 2 2
H_{e_{n}} := (-1)^{n} e^{{x^2} \over {2}} {{d^{n}} \over {dx^{n}}} e^{- {{x^2} \over {2}}}
H e n := ( − 1 ) n e 2 x 2 d x n d n e − 2 x 2
Explanation By definition, H n H_{n} H n ‘function’ , but it is conventionally called Hermite ‘polynomial’ . This is not just in Korean, as the English expression is also Hermite polynomial, not polynomial function.
The first few Hermite polynomials are as follows:
H 0 ( x ) = 1 H 1 ( x ) = 2 x H 2 ( x ) = 4 x 2 − 2 H 3 ( x ) = 8 x 3 − 12 x H 4 ( x ) = 16 x 4 − 48 x 2 + 12 H 5 ( x ) = 32 x 5 − 160 x 3 + 120 x ⋮
\begin{align*}
H^{0}(x) &= 1 \\
H^{1}(x) &= 2x \\
H^{2}(x) &= 4x^{2}-2 \\
H^{3}(x) &= 8x^{3}-12x \\
H^{4}(x) &= 16x^{4}-48x^{2}+12 \\
H^{5}(x) &= 32x^{5}-160x^{3}+120x \\
&\vdots
\end{align*}
H 0 ( x ) H 1 ( x ) H 2 ( x ) H 3 ( x ) H 4 ( x ) H 5 ( x ) = 1 = 2 x = 4 x 2 − 2 = 8 x 3 − 12 x = 16 x 4 − 48 x 2 + 12 = 32 x 5 − 160 x 3 + 120 x ⋮
Properties Orthogonality Hermite polynomials are orthogonal with respect to the weight function w ( x ) = e − x 2 w(x)=e^{-x^{2}} w ( x ) = e − x 2 ( − ∞ , ∞ ) (-\infty, \infty) ( − ∞ , ∞ ) Link )
⟨ H n ∣ H m ⟩ e − x 2 = ∫ − ∞ ∞ e − x 2 H n ( x ) H m ( x ) d x = π 2 n n ! δ n m
\braket{ H_{n} | H_{m} }_{e^{-x^{2}}} =\int_{-\infty}^{\infty}e^{-x^{2}}H_{n}(x)H_{m}(x)dx=\sqrt{\pi}2^{n}n!\delta_{nm}
⟨ H n ∣ H m ⟩ e − x 2 = ∫ − ∞ ∞ e − x 2 H n ( x ) H m ( x ) d x = π 2 n n ! δ nm
Recurrence relation Hermite polynomials satisfy the following recurrence relation. (Link )
H n ′ ( x ) = 2 n H n − 1 ( x ) H n + 1 ( x ) = 2 x H n ( x ) − 2 n H n − 1 ( x ) = 2 x H n ( x ) − H n ′ ( x )
\begin{align*}
H_{n}^{\prime}(x) &= 2nH_{n-1}(x) \\
H_{n+1}(x) &= 2xH_{n}(x)-2nH_{n-1}(x) \\
&= 2xH_{n}(x)-H_{n}^{\prime}(x) \nonumber
\end{align*}
H n ′ ( x ) H n + 1 ( x ) = 2 n H n − 1 ( x ) = 2 x H n ( x ) − 2 n H n − 1 ( x ) = 2 x H n ( x ) − H n ′ ( x )
Generating function The generating function of Hermite polynomials is as follows. (Link )
Φ ( x , t ) = ∑ n = 0 ∞ H n ( x ) n ! t n = e 2 x t − t 2
\Phi (x,t)=\sum \limits _{n=0}^{\infty} \frac{H_{n}(x)}{n!}t^{n}= e^{2xt-t^{2}}
Φ ( x , t ) = n = 0 ∑ ∞ n ! H n ( x ) t n = e 2 x t − t 2
