Modular differential equations and orthogonal polynomials

Abstract

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.