We present new representations of Gauss--Legendre polynomials and their derivatives in the shifted power basis and in bases related to symmetric orthogonal Jacobi polynomials. Using these representations and certain recurrence relations, we propose efficient $O(n^2+dn)$ methods for evaluating a Gauss--Legendre curve of degree $n$ in $\mathbb E^d$. We also propose algorithms for multipoint evaluation with computational complexity $O(Mdn+dn^2)$, where $M$ is the number of evaluation points.

翻译：我们提出了高斯-勒让德多项式及其导数在平移幂基和对称正交雅可比多项式相关基下的新表示。利用这些表示及某些递推关系，我们提出了在 $\mathbb E^d$ 上求值 $n$ 次高斯-勒让德曲线的高效 $O(n^2+dn)$ 方法。我们还提出了多点求值的算法，其计算复杂度为 $O(Mdn+dn^2)$，其中 $M$ 是求值点的数量。