TITLE: Why are the Gegenbauer polynomials called "ultraspherical"?
QUESTION [5 upvotes]: 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?

REPLY [4 votes]: 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.