Special solutions to Chazy equation
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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.
KeywordsChazy equation hypergeometric solution modular group Eisenstein series theta functions sum of divisors Riemann hypothesis
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- 3.G. Halphen, “Sur une systéme d'équations différentielles,” C.R. Acad. Sci. Paris 92, 1101–1103 (1881).Google Scholar
- 11.Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol.1.Google Scholar
- 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.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.Sloane Online Encyclopedia of Integer Sequences. http://oeis.org/wiki/Sum_of_divisors_function.Google Scholar
- 16.C. Carathéodory, Theory of Functions of a Complex Variable (Chelsea, New York, 1954), Vol.2.Google Scholar
- 17.S. Chakravarty and M. J. Ablowitz, Parameterizations of the Chazy Equation. http://arxiv.org/abs/0902.3468v1.Google Scholar
- 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
- 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
- 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
- 28.C. G. J. Jacobi, “Fundamenta Nova Theoriae Functionum Ellipticarum,” in Jacobi’s Gesammelte Werke (Chelsea, New York, 1969).Google Scholar