The most useful polynomials you have never heard of.

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[grafrotz]

These polynomials are the Eigenfunctions of the Harmonic Oszillator in Quantum Mechanics, the n labels the Energy level.
The play an important part on the discovery of QM.

[tomkerruish]

If you multiply these polynomials by exp(-x²/2), you get eigenfunctions of the Fourier transform with eigenvalues i^n. Since they span L²(R), they can be used to define a fractional Fourier transform. (Technically, it's a Plancherel transform, since we're dealing with L², not L¹.)

[emanuellandeholm]

As an audio/DSP nerd I'm really interested in the orthogonal polys. Hermite, Lagrange, Chebychev et alia. They form a solid background for vector space theory and application. So useful.

In particular, the Hermite polys are sometimes used for time domain interpolation in synthesizer applications. Cheap and effective.

[spicymickfool]

I think you can also get those using the Graham-Schmidt orthonormalization process with those starting H_0(x) and H_1(x) and weight of e^(-x^2).

[ketv]

I certainly encountered them - in my undergrad course on functions of a real variable - because they are orthogonal with the metric (f∘g)(x) = ∫ [from -∞ to ∞] f(x) g(*x) exp(-x²) dx - giving an orthogonal basis for the Hilbert space of exponentially bounded (there are weaker conditions) functions over the whole real number line (just as the Fourier transform gives a basis over a finite interval and the Laplace transform gives a basis over the positive reals). (By then, I'd seen it in both an E&M class and a quantum class - the physicists always seem to use mathematics that the mathematicians haven't yet taught.)

The Hermite functions ψₙ(x) = Hₙ(x) exp(-x²/2)/sqrt(2ⁿ n! sqrt(π)) are the eigenstates of the quantum harmonic oscillator, and of course they turn up all over probability theory. They're the eigenfunctions of the continuous Fourier transform. They underlie wavelet analysis.

By the way, Charles Hermite was French, and pronounced his name more like "sharl air-MEET"

[coreyyanofsky]

"Hermite" is a French name and is pronounced approximately like "air-meet"

[ianmathwiz]

I've heard of them before, in the context of the quantum harmonic oscillator.

[trueriver]

I've never before stopped to think of the constant 1 being a polynomial...

[davidgillies]

Hermite polynomials were the first orthonormal polynomials we studied in introductory quantum mechanics in the first year of my physics degree (followed by Legendre, Chebyshev, Laguerre etc.).

[xizarrg]

Sure would be nice for future searchability if you included the type of polynomial in the title. (e.g., "Hermite Poly'ls: the most useful poly'ls you've never heard of".)

[jamesfortune]

I believe they came up in my Numerical Analysis class. There weren't many math surprises in my engineering classes. 😊

[Calcprof]

I've heard of them. Parkly as an example of the three term recurrence theorem for orthogonal polynomials in my graduate applied analysis course (that is the course that I taught for many years) Great polynomials. And of course on PDE courses.

[toddtrimble]

You're absolutely right, I never before tonight heard of the "her might" polynomials.

[lucasf.v.n.]

good video; I wish u had shown the applications in physics;

[ethanbartiromo]

I didn’t recognize it until the competing definition, I remembered doing something similar in my numerical analysis class, but then I explicitly remember the alternative definition in my numerical analysis class

[notanotherraptor]

Glad to see the magic checkboxes are back!

[alexeyvlasenko]

I've definitely heard of Hermite polynomials. They come up very naturally in quantum mechanics.

[funatish]

As a physicist who never used Hermite Polynomials after undergrad, this is triggering my PTSD of Mathematical Physics and Quantum Mechanics classes.

[hgh]

It's cool to see a function expressed like matrix diagonalization.

[topquark]

I happened to see in the street
The mathematician Hermite.
After I said "Bonjour"
He replied, "Oui, bien sûr,
I am happy with you so to meet."