We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets us give a simple and unified view of previous algorithms computing these recurrences, with a noncommutative Euclidean algorithm as the algorithmic engine. Finally, we demonstrate the effectiveness of our approach on various examples.

翻译：本文考虑经典正交多项式基下的级数展开。当此类级数求解具有多项式系数的线性微分方程时，其系数满足线性递归方程。我们将该方程解释为线性递归算子分数的分子。这一解释使我们能够以非交换欧几里得算法为算法引擎，为先前计算这些递归的算法提供统一且简洁的视角。最后，我们通过多个实例展示了该方法的有效性。