On modular solutions of certain meromorphic modular differential equations
type: Item
instance of: scholarly article
title: On modular solutions of certain meromorphic modular differential equations
zbMATH Open document ID: 1256.11031
author: Yuichi Sakai
published in: The Ramanujan Journal
publication date: 2010-09-20T00:00:00Z
review text: One of the differential equations under consideration reads \[ f''- \Biggl({k+1\over 6} E_2-{E_6\over 3E_4}\Biggr)\,f'+ \Biggl({k(k+1)\over 12}\,E_2'- {k\over 18} {E_6'\over E_4}\Biggr)\,f= 0, \] where \(E_2\), \(E_4\), \(E_6\) are Eisenstein series and where the parameter \(k\) can take positive integers or certain positive rational numbers with denominators 2 or 5 as its values. The author gives solutions of this differential equation in terms of modular forms of weight \(k\) and levels \(\leq 5\) which also involve Gaussian hypergeometric series and the so-called Heun local series. The reader will find definitions of several interesting modular forms and functions of levels \(\leq 5\). Similar results for another (somewhat simpler) differential equation were obtained by \textit{M. Kaneko} and \textit{M. Koike} [Ramanujan J. 7, No. 1--3, 145--164 (2003; Zbl 1050.11047)].
reviewed by: Günter Köhler
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F37
Mathematics Subject Classification ID: 11F03
zbMATH DE Number: 5788048
zbMATH Keywords: modular form
zbMATH Keywords: quasimodular form
zbMATH Keywords: differential equations
MaRDI profile type: Publication
full work available at URL: s11139-010-9236-3
OpenAlex ID: W2076122552
cites work: On Modular forms of Weight (6n + 1)/5 Satisfying a Certain Differential Equation
cites work: On modular forms arising from a differential equation of hypergeometric type
cites work: ON EXTREMAL QUASIMODULAR FORMS
cites work: Q4857337
cites work: Modular differential equations of second order with regular singularities at elliptic points for $SL_2(\mathbb {Z})$
DOI: 10.1007/S11139-010-9236-3
description: scientific article; zbMATH DE number 5788048