标题： 模微分方程和正交多项式 显示英文标题

标题： Modular differential equations and orthogonal polynomials

摘要： 我们研究解在模群下等变变换的二阶模微分方程。 在可约情况下，我们使用涉及埃森斯坦级数和$J$-不变量的显式假设构造所有此类解，将问题简化为一个代数系统。 我们证明该系统的根由满足Fuchs微分方程的正交多项式捕获。 它们的递推关系、范数和权函数被推导出来，完成了在此设置下等变解的分类。 显示英文摘要

摘要： We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the $J$-invariant, reducing the problem to an algebraic system. We show that the roots of this system are captured by orthogonal polynomials satisfying a Fuchsian differential equation. Their recurrence, norms, and weight function are derived, completing the classification of equivariant solutions in this setting.