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Faà di Bruno’s Formula and Modular Forms

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Abstract

We show that Faà di Bruno’s formula can play important roles in modular forms theory and in the study of differential operators of the form \( \displaystyle \left( a(x)\frac{d}{dx} \right) ^n\). We also emphasize the importance of the fundamental forms \(\displaystyle y_k= \Delta ^{-\frac{k}{12}}, \Delta \) is the discriminant function, making a link between some aspects of differential Galois theory and modular forms.

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References

  1. Appell, P.: Mémoire sur les équations différentielles linéaires. Annales Scientifiques de l’ENS, \(2^{e}\) série, tome 10, 391–424 (1881)

  2. Berndt, B.C., Chan, H.H., Huang, S.S.: Incomplete Elliptic Integrals in Ramanujan’s Lost Notebook in q-series from a Contemporary Perspective, pp. 79–126. In: Ismail, M.E.H., Stanton, D. (eds.) Amer. Math. Soc., USA (2000)

    Google Scholar 

  3. Bol, G.: Invarianten linearer differentialgleichungen. Abh. Math. Sem. Univ. Hamburger Univ. 16, 1–28 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bronstein, M., Mulders, T., Weil, J-A.: On symmetric powers of differential operators. In: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation. ACM Press, New York (1997)

  5. Fricke, R., Klein, F.: Vorlesungen über die theorie der elliptischen modulfunctionen. Teubner, Leipzig (1890)

    MATH  Google Scholar 

  6. Gustafsson, G., Peetre, J.: Notes on projective structures on complex manifolds. Nagoya Math. J. 116, 63–88 (1989)

    MathSciNet  MATH  Google Scholar 

  7. Hardy, G.H.: On two theorems of F. Carlson and S. Wigert. Acta Mathematica 42, 327–339 (1920)

    Article  Google Scholar 

  8. Hurwitz, A.: Über die Differentsialglechungen dritter ordnung, welchen die Formen mit linearen transformationen in sich genügen. Math. Ann. 36, 97–112 (1848). (Ges. math. Werke, Bd. 2, 171–191)

  9. Johnson, W.P.: The curious history of Faà di Bruno’s formula. Am. Math. Monthly 109, 217–234 (2002)

    Article  MATH  Google Scholar 

  10. Lang, S.: Introduction to Modular Forms. A Series of Comprehensive Studies in Mathematics, vol. 222. Springer, Berlin (1976)

  11. 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, 261–272, 272–284 (1981). (Indag. Math. 13, 1951)

  12. Van der Put, M., Singer, M.F.: Galois Theory of Linear Differential Equations Grundlehren der Mathematischen Wissenschaften, vol. 328. Springer, Berlin (2003)

    Google Scholar 

  13. Rankin, R.A.: The construction of automorphic forms from the derivatives of given forms. J. Indian Math. Soc. 20, 103–116 (1956)

    MathSciNet  MATH  Google Scholar 

  14. Rankin, R.A.: The construction of automorphic forms from the derivatives of given forms. Mich. Math. J. 4, 181–186 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  15. Rademacher, H.: Topics in Analytic Number Theory. Die Grundlehren der Math. Wissenschaften, Band 169. Springer, Berlin (1973)

  16. Resnikoff, H.L.: On differential operators and automorphic forms. Trans. Am. Math. Soc. 124, 334–346 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sebbar, Ab: Sebbar, Ah: Eisenstein series and modular differential equations. Can. Math. Bull. 55(2), 400–409 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Serre, J.P.: Cours d’Arithmétique. Presses Universitaires de France, France (1970)

    MATH  Google Scholar 

  19. Singer, M.F.: Solving homogeneous linear differential equations in terms of second order linear differential equations. Am. J. Math. 107, 663–696 (1985)

    Article  MATH  Google Scholar 

  20. Touchard, J., Van Der Pol, B.: Equations différentielles linéaires vérifiées par certaines fonctions modulaires elliptiques. Nederl. Akad. Wetensch. Proc. Ser. A. 59=. Indag. Math. 18, 166–169 (1956)

    Article  Google Scholar 

  21. Wilczynski, E.J.: Projective Differential Geometry of Curves and Ruled Surfaces. Teubner (1906). (reprinted by Chelsea Publ. Co.)

  22. Zagier, D.: Elliptic modular forms and their applications. In: The 1-2-3 of Modular Forms. Universitext. Springer, Berlin, pp. 1–103 (2008)

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Authors and Affiliations

  1. Ecole Nationale Supéprieure de Technologie, Route Nationale N 5, ZI, Rouiba-Alger, Algeria

    Djohra Meguedmi

  2. IMB, Institut de Mathématiques de Bordeaux, Université Bordeaux I, 351 Cours de la Libération, 33405, Talence, France

    Ahmed Sebbar

Authors
  1. Djohra Meguedmi
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  2. Ahmed Sebbar
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Correspondence to Ahmed Sebbar.

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Communicated by Daniel Aron Alpay.

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Meguedmi, D., Sebbar, A. Faà di Bruno’s Formula and Modular Forms. Complex Anal. Oper. Theory 10, 409–435 (2016). https://doi.org/10.1007/s11785-015-0494-3

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