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

$\displaystyle Q_n(x) = (1 - x^2)P_n'(x)$

are orthogonal on $\displaystyle [-1,1]$ with respect to the weight function $\displaystyle (1-x^2)^{-1}$ 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?