Modular differential equations and orthogonal polynomials

TL;DR

This paper classifies solutions to second-order modular differential equations that transform under the modular group, linking them to orthogonal polynomials and providing explicit constructions and properties.

Contribution

It introduces an explicit ansatz involving Eisenstein series and the J-invariant to construct solutions and connects these to orthogonal polynomials satisfying Fuchsian equations.

Findings

Solutions are explicitly constructed using Eisenstein series and J-invariant.

Roots of algebraic systems correspond to orthogonal polynomials with specific properties.

Complete classification of equivariant solutions in the reducible case.

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.