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Hermite Polynomials

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Hermite polynomials are a widely used family of weight function proportional to w(x) = e-x2.

Definition of Hermite Polynomials

There are several definitions for “Hermite polynomials”, which can be a source of confusion. First, two different starting points result in two different sets of polynomials, often called the “physicists” and “probabilists'” polynomials. Most authors simply refer to Hermite polynomials without any clarification, assuming the reader is working in one field or another (i.e. physics or probability) and therefore don’t need to know the “other” definition.


If you’re in calculus, you’re likely dealing with the “physicists” Hermite polynomials, built from the monomials. The first few are (Sawitzki, 2009):

  • H0 (x) = 1
  • H1 (x) = x
  • H2 (x) = x2 – 1
  • H3 (x) = x3 – 3x
  • H4 (x) = x4 – 6x2 + 3
  • H5 (x) = x5 – 10x3 + 15x
  • H6 (x) = x6 – 15x4 + 45x2 – 15
hermite polynomials

The first four physicists Hermite polynomials (graphed at Desmos.com)

An alternate definition, with w(x) = e-x2/2 is sometimes used, especially in statistics. The “probabilists'” polynomials are sometimes called Chebyshev-Hermite polynomials (Sawitzki, 2009). The first few are:


  • H0 (x) = 1
  • H1 (x) = 2x
  • H2 (x) = 4x2 – 2
  • H3 (x) = 8x3 – 12x
  • H4 (x) = 16x4 – 48x2 + 12
  • H5 (x) = 32x5 – 160x3 + 120x
  • H6 (x) = 64x6 – 480x4 + 720x2 – 120

Hermite Interpolation and Other Uses

Hermite polynomials are very useful as interpolation functions because their value—and their derivatives values— up to order n are unity at zero at the endpoints of the closed interval [0, 1] (Huebner et al., 2001). They provide an alternative way of representing cubic curves, allowing the curve to be defined in terms of endpoints and derivatives at those endpoints (Buss, 2003).

Hermite polynomials occur in various areas of physics, including as part of the solution to the quantum harmonic oscillator Hamiltonian. They also arise in numerical analysis as Gaussian quadrature.

References

Buss, S. (2003). 3D Computer Graphics. A Mathematical Introduction with OpenGL. Cambridge University Press.
Desmos Graphing Calculator.
Huebner, K. et al. (2001). The Finite Element Method for Engineers. Wiley.
Sawitzki, G. (2009). Computational Statistics: An Introduction to R, CRC Press.

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Stephanie Glen. "Hermite Polynomials" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/hermite-polynomials/
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