Why are the Gegenbauer polynomials called "ultraspherical"?

Question

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?

Answer 1

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 $C_n^{\frac{3}{2}-1}(x)=P_n(x)$.

Answer 2

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.