Why are the Gegenbauer polynomials called "ultraspherical"?

There needs to be an excellent reason that the Gegenbauer polynomials were additionally called "ultraspherical" polynomials. I realize that when $\alpha=\frac{1}{2}$, the Gegenbauer polynomials lower to the Legendre polynomials, and also the Legendre polynomials are made use of in specifying Spherical harmonics. Yet that is regarding I recognize just how to take that thinking.

Exists a visualization of these polynomials that fits on a round? What is an ultrasphere anyhow?

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2022-07-25 20:41:26
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