Special solutions to Chazy equation

Article

Abstract

We consider the classical Chazy equation, which is known to be integrable in hypergeometric functions. But this solution has remained purely existential and was never used numerically. We give explicit formulas for hypergeometric solutions in terms of initial data. A special solution was found in the upper half plane H with the same tessellation of H as that of the modular group. This allowed us to derive some new identities for the Eisenstein series. We constructed a special solution in the unit disk and gave an explicit description of singularities on its natural boundary. A global solution to Chazy equation in elliptic and theta functions was found that allows parametrization of an arbitrary solution to Chazy equation. The results have applications to analytic number theory.

Keywords

Chazy equation hypergeometric solution modular group Eisenstein series theta functions sum of divisors Riemann hypothesis 

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References

  1. 1.
    B. C. Berndt, Number Theory in the Spirit of Ramanujan (Am. Math. Soc., Providence, 2006).CrossRefMATHGoogle Scholar
  2. 2.
    J. Chazy, “Sur les équations différentielles dont l’intégrale générale est uniforme et admet des singularitiés essentielles mobiles,” C.R. Acad. Sci. Paris 149, 563–565 (1909).MATHGoogle Scholar
  3. 3.
    G. Halphen, “Sur une systéme d'équations différentielles,” C.R. Acad. Sci. Paris 92, 1101–1103 (1881).Google Scholar
  4. 4.
    M. J. Ablowitz, S. Chakravarty, and H. Halm, “Integrable systems and modular forms of level 2,” J. Phys. A: Math. Gen. 39, 15341–15353 (2006).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    P. A. Glarkson and P. J. Olver, “Symmetry and the Chazy equation,” J. Differ. Equations 124, 225–246 (1996).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    H. Blasius, “Grenzschichten in Flüssigkeiten mit kleiner Reibung,” Z. Math. Phys. 56, 1–37 (1908).MATHGoogle Scholar
  7. 7.
    J. P. Boyd, “The Blasius function in the complex plane,” Experiment. Math. 8, 381–394 (1999).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    V. P. Varin, “A solution of the Blasius problem,” Comput. Math. Math. Phys. 54 (6), 1025–1036 (2014).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and the Inverse Scattering, Lect. Notes Math., Vol. 149 (Cambridge Univ. Press, Cambridge, 1991).CrossRefMATHGoogle Scholar
  10. 10.
    Z. Nehari, Conformal Mapping (McGraw-Hill, New York, 1952).MATHGoogle Scholar
  11. 11.
    Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol.1.Google Scholar
  12. 12.
    T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory (Springer-Verlag, New York, 1990).CrossRefMATHGoogle Scholar
  13. 13.
    N. Joshi and M. D. Kruskal, “A local asymptotic method of seeing the natural barrier of the solutions of the Chazy equation,” in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, Ed. by P. A. Clarkson, NATO ASI Ser. C: Math. Phys. Sci., Vol. 413 (Kluwer, Dordrecht, 1992).Google Scholar
  14. 14.
    M. D. Kruskal, N. Joshi, and R. Halburd, “Analytic and asymptotic methods for nonlinear singularity analysis: A review and extensions of tests for the Painlevé property,” in Integrability of Nonlinear Systems, Ed. by Y. Kosmann-Schwarzbach (Springer, Berlin, 2004).Google Scholar
  15. 15.
    Sloane Online Encyclopedia of Integer Sequences. http://oeis.org/wiki/Sum_of_divisors_function.Google Scholar
  16. 16.
    C. Carathéodory, Theory of Functions of a Complex Variable (Chelsea, New York, 1954), Vol.2.Google Scholar
  17. 17.
    S. Chakravarty and M. J. Ablowitz, Parameterizations of the Chazy Equation. http://arxiv.org/abs/0902.3468v1.Google Scholar
  18. 18.
    D. Zagier, “Elliptic modular forms and their applications”, in The 1-2-3 on Modular Forms, Ed. by J. H. Bruinier et al. (Springer, Berlin, 2008).Google Scholar
  19. 19.
    M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).MATHGoogle Scholar
  20. 20.
    S. Ramanujan, “On certain arithmetical functions,” Trans. Camb. Philos. Soc. 22, 159–184 (1916); in Collected Papers of Srinivasa Ramanujan, Ed. by G. H. Hardy et al. (Cambridge Univ. Press, Cambridge, 1927).Google Scholar
  21. 21.
    P. C. Toh, “Differential equations satisfied by Eisenstein series of level 2,” Ramanujan J. 25, 179–194 (2011).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    J. G. Huard et al., “Elementary evaluation of certain convolution sums involving divisor functions,” in Number Theory for the Millennium II, Ed. by M. A. Bennett (A. K. Peters, Natick, Mass., 2002), pp. 229–274.Google Scholar
  23. 23.
    J. C. Lagarias, “An elementary problem equivalent to the Riemann hypothesis,” Math. Mon. 109 (6), 534–543 (2002).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    V. P. Varin, “Flat expansions and their applications,” Comput. Math. Math. Phys. 55 (5), 797–810 (2015).MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Y. V. Nesterenko, “Algebraic independence for values of Ramanujan functions,” in Introduction to Algebraic Independence Theory, Ed. by Y. V. Nesterenko and P. Philippon (Springer, Berlin, 2001).CrossRefGoogle Scholar
  26. 26.
    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, New York, 2007).MATHGoogle Scholar
  27. 27.
    E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge Univ. Press, Cambridge, 1927).MATHGoogle Scholar
  28. 28.
    C. G. J. Jacobi, “Fundamenta Nova Theoriae Functionum Ellipticarum,” in Jacobi’s Gesammelte Werke (Chelsea, New York, 1969).Google Scholar
  29. 29.
    G. Robin, “Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann,” J. Math. Pures Appl. Neuv. Ser. 63 (2), 187–213 (1984).MATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Keldysh Institute of Applied MathematicsRussian Academy of SciencesMoscowRussia

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