Shifted Legendre polynomials

Summary

The shifted Legendre polynomials are the Legendre polynomials moved from the interval $[-1, 1]$ to the interval $[0, 1]$. They are given by $\tilde{P}_n(x) = P_n(2x - 1)$. They are orthogonal on $[0, 1]$ for the constant weight. So they are the natural polynomial basis for least squares and Gaussian quadrature on the unit interval.