Playing around for a solver for orthogonal polynomials in differential equations I stumbled upon the Gegenbauer polynomials described on Wikipedia the other day. They are the family of polynomials $y(x)$ satisfying the following differential equation: $$(1-x^2)y'' - (2\alpha+1) xy' +n(n+2\alpha)=0$$
I am curious in which applications they show up. In particular the one where $\alpha = 3/2$ which seems to correspond to the constant polynomial have it's function become a semi-circle. The function being constructed as : $$f_k(x) = \sqrt{w(x)}\cdot P_k(x)$$
In other words, building the functions so that they become orthogonal under the trivial (weight=1) scalar product, because
$$<f_i,f_j> = \int_{-1}^1 f_i(x)f_j(x)dx = \int_{-1}^1 P_i(x)P_j(x)w(x)dx = \delta_{i-j}$$
and since $w(x) = (1-x^2)^{\alpha-1/2}$, for $\alpha=3/2$ we get
$$\sqrt{w(x)} = (1-x^2)^{(3/2-1/2)/2} = (1-x^2)^{1/2}$$ which is the curve of a circle (in fact the unit circle centered on origo) we can see in the plot for the function corresponding to the constant (bleu) polynomial below.
In what kind of "real-world" application do these show up? Any branch of science or engineering is of interest.
Gegenbauer are generalizations to the Legendre Polynomials, this is easy to see from their generating function:
$$ \frac{1}{(1 - 2xt + t^2)^{\alpha}} = \sum_{n=0}^{+\infty}C_n^{(\alpha)}t^n $$
which should be compared with the generating function for Legendre polynomials
$$ \frac{1}{(1 - 2xt + t^2)^{1/2}} = \sum_{n=0}^{+\infty}P_n(x)t^n $$
As such Gegenbauer polynomials provide a natural extension to spherical harmonics in higher dymensions: hyperspherical harmonics. In physics the set $P_n(x)$ is used to assemble the basis for solving the Hydrogen atom (two body problem), it is but expected that the set $C_n^{(\alpha)}$ can be used to solve a Coulomb problem in higher dimensions: and they do!
They are used for describing a basis for a many problem. Have a look at this reference.