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?