I am looking for an othogonal basis of functions over the interval $[0,1]$ with weight function $(1-x^2)^{\alpha-1/2}$. Gegenbauer polynomials are frustratingly close to what I need, but they are defined over the interval $[-1,1]$, and a change of variables ends up changing the weight function.

None of the orthogonal polynomial families I have looked at (Chebyshev, Gegenbauer, Legendre, Laguerre, Jacobi, Hermite) have this property.

Does anyone know of a family that does? Suggestions for references that may point the way would also be very helpful.

Thanks!