The Legendre polynomials, , which are orthogonal on the interval , are useful for spectral methods. However they do not satisfy any conditions like , which are desirable for some boundary value problems. The polynomials defined by
are orthogonal on with respect to the weight function 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?