The Legendre polynomials, P_n, which are orthogonal on the interval [-1, 1], are useful for spectral methods. However they do not satisfy any conditions like P_n(\pm 1) = 0, which are desirable for some boundary value problems. The polynomials defined by

Q_n(x) = (1 - x^2)P_n'(x)

are orthogonal on [-1,1] with respect to the weight function (1-x^2)^{-1} and they satisfy homogeneous boundary conditions. This class of polynomials is mentioned in Quateroni and Valli, Numerical Approximation of Partial Differential Equations (Springer 2008).

Does anyone know if they have a special name?