Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

There has to be a good reason why the Gegenbauer polynomials were also named "ultraspherical" polynomials. I am aware that when $\alpha=\frac{1}{2}$, the Gegenbauer polynomials reduce to the Legendre polynomials, and the Legendre polynomials are used in defining Spherical harmonics. But that is as far as I know how to take that reasoning.

Is there a visualization of these polynomials that fits on a sphere? What is an ultrasphere anyway?

share|improve this question

2 Answers 2

up vote 3 down vote accepted

I am aware that when $\alpha=\frac12$, the Gegenbauer polynomials reduce to the Legendre polynomials, and the Legendre polynomials are used in defining spherical harmonics.

You pretty much nailed it. From here:

In the theory of hyperspherical harmonics, Gegenbauer polynomials play a role which is analogous to the role played by Legendre polynomials in the theory of the familiar three-dimensional spherical harmonics.

As a compressed version of the discussion in the book (look there for more details), the ultraspherical/hyperspherical harmonics involve Gegenbauer polynomials of the form $C_n^{\frac{d}{2}-1}(x)$, where $d$ is the dimension of the hyperspherical harmonics being considered. For the usual case of $d=3$, we have $C_n^{\frac{3}{2}-1}(x)=P_n(x)$.

share|improve this answer

Legendre polynomials arise when solving 3D Laplace's equation in spherical coordinates by separation of variables, i.e. $f(x,y,z) = f_1(r) Y(\theta, \phi)$.

Legendre polynomials appear in spherical harmonics, specifically for $\ell \in \mathbb{Z}_{\geqslant 0}$ and $m\in \mathbb{Z}$, such that $-\ell \leqslant m \leqslant \ell$: $$ Y_{\ell,m}(\theta, \phi) = \sqrt{\frac{2 \ell+1}{4 \pi}} \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} P_\ell^m(\cos \theta) \mathrm{e}^{i m \phi} $$

Solving Laplace's equation in $\mathbb{R}^d$ in hyper-spherical coordinates (in older literature a.k.a ultra-spherical coordinates) by separation of variables, gives rise to ultraspherical polynomials.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.