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Eisenstein Series and Modular Differential Equations

Published online by Cambridge University Press:  20 November 2018

Abdellah Sebbar  and
Ahmed Sebbar
Abdellah Sebbar
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5e-mail: asebbar@uottawa.ca
Ahmed Sebbar
Affiliation:
Institut de Mathématiques de Bordeaux, Université Bordeaux 1 351, cours de la Libération F-33405 Talence cedexe-mail: ahmed.sebbar@math.u-bordeaux1.fr
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Abstract

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The purpose of this paper is to solve various differential equations having Eisenstein series as coefficients using various tools and techniques. The solutions are given in terms of modular forms, modular functions, and equivariant forms.

Keywords

11F1134M05differential equationsmodular formsSchwarz derivativeequivariant forms
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 2 , 01 June 2012 , pp. 400 - 409
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Brady, M. M., Meromorphic solutions of a system of functional equations involving the modular group. Proc. Amer.Math. Soc. 30(1971), 271–277. http://dx.doi.org/10.1090/S0002-9939-1971-0280712-5 Google Scholar
[2] Hurwitz, A., Ueber die Differentialgleichungen dritter Ordnung, welchen die Formen mit linearen Transformationen in sich genÜgen. Math. Ann. 33(1889), no. 3, 345–352. http://dx.doi.org/10.1007/BF01443965 Google Scholar
[3] Kaneko, M., and Koike, M., On modular forms arising from a differential equation of hypergeometric type. Ramanujan J. 7(2003), no. 1-3, 145–164. http://dx.doi.org/10.1023/A:1026291027692 Google Scholar
[4] Klein, F., Ueber Multiplicatorgleichungen. Math. Ann. 15(1879), no. 1, 86–88. http://dx.doi.org/10.1007/BF01444105 Google Scholar
[5] Knopp, M., Rational period functions of the modular group. Duke Math J. 45(1978), no. 1, 47–62. http://dx.doi.org/10.1215/S0012-7094-78-04504-0 Google Scholar
[6] McKay, J. and Sebbar, A., Fuchsian groups, automorphic functions and Schwarzians. Math. Ann. 318(2000), no. 2, 255–275. http://dx.doi.org/10.1007/s002080000116 Google Scholar
[7] Mathur, S., Mukhi, S., and Sen, A., On the classifiation of rational conformal field theories. Phys. Lett. B 213(1988), no. 3, 303–308. http://dx.doi.org/10.1016/0370-2693(88)91765-0 Google Scholar
[8] Milas, A., Ramanujan's “Lost Notebook” and the Virasoro algebra. Comm. Math. Phys. 251(2004), no. 3, 657–678. http://dx.doi.org/10.1007/s00220-004-1179-3 Google Scholar
[9] Ramanujan, S., On certain arithmetical functions. Trans. Cambridge Philos. Soc. 22(1916), 159–184.Google Scholar
[10] Schwarz, H., Gesammelte Mathmatische Abhandlungen. Vol. 2, Berlin, 1880.Google Scholar
[11] Sebbar, A., Torsion-free genus zero congruence subgroups of PSL 2(R) . Duke Math. J. 110(2001), no. 2, 377–396. http://dx.doi.org/10.1215/S0012-7094-01-11028-4 Google Scholar
[12] Sebbar, A. and Sebbar, A., Equivariant functions and integrals of elliptic functions. http://mysite.science.uottawa.ca/asebbar/publi/equivariant-elliptic.pdf Google Scholar
[13] Smart, J. R., On meromorphic functions commuting with elements of a function group. Proc. Amer. Math. Soc. 33(1972), 343–348. http://dx.doi.org/10.1090/S0002-9939-1972-0293086-1 Google Scholar
[14] Touchard, J., and Van der Pol, B., Equations différentielles linéaires vérifiées par certaines fonctions modulaires elliptiques. Nederl. Akad.Wetensch. Proc. Ser. A. 59 = Indag. Math. 18(1956), 166–169.Google Scholar
[15] Van der Pol, B., On a non-linear partial differential equation satisfied by the logarithm of the Jacobian theta-functions, with arithmetical applications. I, II. Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indag. Math. 13(1951), 261–271, 272–284.Google Scholar