Gegenbauer transform

Summary

The Gegenbauer transform is a finite integral transform on $[-1, 1]$. It sends a function to its integrals against the Gegenbauer polynomials $C_n^{\lambda}$ with the weight $(1 - x^2)^{\lambda - 1/2}$. It is the symmetric case $\alpha = \beta$ of the Jacobi transform, up to constant factors. It turns the Gegenbauer differential operator into multiplication by $-n(n + 2\lambda)$. For $\lambda = 1/2$ it is the Legendre transform.