Ultraspherical

When I hear the term ultraspherical I think of something extremely spherical. For example, a baseball is spherical, but a billiard ball is more spherical. Maybe a highly polished billiard ball is ultraspherical. Using this line of thought, the term ultraspherical polynomial is inexplicable. This is an example of the arcane terminology I wrote about recently. In this post I’ll explain what it conveys. A spherical polynomial is a polynomial that naturally falls out of solving Laplace’s equation in spherical coordinates, using separation of variables. Legendre polynomials are spherical polynomials. Gegenbauer polynomials are so called because a man named Gegenbauer studied them, just as Legendre polynomials take their name from Legendre. Gegenbauer polynomials are also called ultraspherical polynomials. Why is that? There are two possible reasons. I’m not sure which is the historical reason, but both are plausible and are useful mnemonics. Ultraspherical polynomials are not extremely…