On modular solutions of certain meromorphic modular differential equations (Q993489)

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)].