Hermite Polynomials

• The Hermite polynomials $H_k(x)$ are solutions to the Hermite differential equation of the form

  $$a(x)y''+b(x)y'+c(x)y=0$$

  where $a(x)=1$, $b(x)=-2x$ and $c(x)=2k$ (positive integer parameter k)

  $$\boxed{\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+2ky=0}$$

• $y_k$ is a solution of the Hermite equation. Therefore, defining $H^k(x) = y_k$.
• A natural one to define Hermite polynomials is through the so-called Rodrigues'formula:

  $$\boxed{H_k(x)=(-1)^ke^{x^2}\frac{d^k}{dx^k}\left[e^{-x^2} \right] }$$

• The first few Hermite polynomials are given by
  

  $$H_0 = 1$$

  $$H_1 = 2 x$$

  $$H_2 = 4 x^{2} - 2$$

  $$H_3 = 8 x^{3} - 12 x$$

  $$H_4 = 16 x^{4} - 48 x^{2} + 12$$

  $$H_5 = 32 x^{5} - 160 x^{3} + 120 x$$

  $$H_6 = 64 x^{6} - 480 x^{4} + 720 x^{2} - 120$$

  
  
• Recursion relation:

  $$\boxed{H_{k+1}(x)=2xH_k(x)-2k H_{k-1}(x)}$$

Example py-file: The program to find first 6 Hermite polynomials: myHermite.py

Figure 5.18: First 6 Hermite Polynomials $H_{k}(x)$ with Recursion Relation: $H_{k+1}(x)=2xH_k(x)-2k H_{k-1}(x)$.

Figure 5.19: Plot of first 6 $H_{k}(x)$.