Hypergeometric solutions to Schwarzian equations
type: Item
instance of: scholarly article
title: Hypergeometric solutions to Schwarzian equations
author: Khalil Besrour
author: Abdellah Sebbar
published in: The Ramanujan Journal
publication date: 2024-11-01T00:00:00Z
review text: Modular forms have a great application area in various parts of mathematics. So there is no surprise that there exist some problems dealing with differential equations. On the other hand, it is well known that Eisenstein series are ``the most natural'' examples of modular forms. When we combine these two, we can consider the following modular differential equation: \N\[\Ny'' +s E_4 y=0 ,\N\]\Nwhere \(E_4\) is weight \(4\) Eisenstein series and \(s=\pi^2 r^2\) with \(r=n/m\) being a rational number in reduced form such that \(m \geq 7\). Furthermore, we can associate this differential equation with a Schwarzian equation as \N\[\N\{ h, \tau \}=2s E_4.\N\]\NThen in the paper under review, the authors solve this Schwarzian equation in terms of the Gauss hypergeometric series by using the theory of equivariant functions on the upper half-plane and the \(2\)-dimensional vector-valued modular forms. This work can be considered as a completion work of a previous one by one of the authors.
reviewed by: İlker İnam
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 34M05
zbMATH DE Number: 7937717
zbMATH Keywords: Eisenstein series
zbMATH Keywords: modular forms
zbMATH Keywords: modular differential equations
zbMATH Keywords: Schwarz derivative
zbMATH Keywords: representations of the modular group
zbMATH Keywords: equivariant functions
MaRDI profile type: Publication
cites work: Rational equivariant forms
cites work: Hypergeometric series, modular linear differential equations and vector-valued modular forms
cites work: On modular forms arising from a differential equation of hypergeometric type
cites work: Q4213385
cites work: VECTOR-VALUED MODULAR FORMS AND LINEAR DIFFERENTIAL OPERATORS
cites work: 2-dimensional vector-valued modular forms
cites work: Fuchsian groups, automorphic functions and Schwarzians
cites work: Automorphic Schwarzian equations
cites work: Automorphic Schwarzian equations and integrals of weight 2 forms
cites work: Equivariant solutions to modular Schwarzian equations
cites work: Modular differential equations and algebraic systems
DOI: 10.1007/S11139-024-00930-6
description: scientific article; zbMATH DE number 7937717