Modular differential equations and orthogonal polynomials

2025/08/14 by Besrour, Khalil, Saber, Hicham, Sebbar, Abdellah
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.10788

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.

Citations

• Hypergeometric solutions to Schwarzian equations
• Modular differential equations and algebraic systems
• Equivariant solutions to modular Schwarzian equations
• Automorphic Schwarzian equations and integrals of weight 2 forms
• Automorphic Schwarzian equations
• On modular forms arising from a differential equation of hypergeometric type

Related

• Equivariant solutions to modular Schwarzian equations
• Modular differential equations and algebraic systems
• On level 2 Modular differential equations
• Differential systems with Fuchsian linear part: correction and linearization, normal forms and multiple orthogonal polynomials
• Defining differential equations for modular forms and Jacobi forms