Special solutions to Chazy equation

V. P. Varin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 2, pp. 210-236

Résumé

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.

Bibliographie

[1] Berndt B. C., Number theory in the spirit of Ramanujan, AMS, Providence, 2006 | MR | Zbl

[2] Chazy J., “Sur les equations differentielles dont l'integrale generale est uniforme et admet des singularities essentielles mobiles”, C.R. Acad. Sc. Paris, 149 (1909), 563–565

[3] Halphen G., “Sur une systeme d'equations differentielles”, C.R. Acad. Sc. Paris, 92 (1881), 1101–1103

[4] Ablowitz M. J., Chakravarty S., Halm H., “Integrable systems and modular forms of level 2”, J. Phys. A: Math. Gen., 39 (2006), 15341–15353 | DOI | MR | Zbl

[5] Glarkson P. A., Olver P. J., “Symmetry and the Chazy equation”, J. of Diff. Eq., 124 (1996), 225–246 | DOI | MR

[6] Blasius H., “Grenzschichten in Flussigkeiten mit kleiner Reibung”, Z. Math. Phys., 56 (1908), 1–37

[7] Boyd J. P., “The Blasius function in the complex plane”, Experiment. Math., 8 (1999), 381–394 | DOI | MR | Zbl

[8] Varin V. P., “A solution of the Blasius problem”, Comput. Math. and Math. Phys., 54:6 (2014), 1025–1036 | DOI | MR | Zbl

[9] Ablowitz M. J., Clarkson P. A., Solitons, nonlinear evolution equations and the inverse scattering, Lect. Notes in Math., 149, C.U.P., Cambridge, 1991 | MR

[10] Nehari Z., Conformal mapping, McGraw-Hill, New York, 1952 | MR | Zbl

[11] Bateman H., Erdelyi A., Higher transcendental functions, v. 1, McGraw-Hill, New York, 1953

[12] Apostol T. M., Modular functions and Diriclilet series in number theory, Springer, New York, 1990 | MR

[13] Joshi N., Kruskal M. D., “A local asymptotic method of seeing the natural barrier of the solutions of the Chazy equation”, Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, NATO ASI Series C: Math, and Phys. Sci., 413, ed. Clarkson P. A., Kluwer, Dordrecht, 1992 | MR

[14] Kruskal M. D., Joshi N., Halburd R., “Analytic and asymptotic methods for nonlinear singularity analysis: A review and extensions of tests for the Painleve property”, Integrability of nonlinear systems, eds. Kosmann-Schwarzbach Y. et al., Springer, Berlin, 2004 | MR | Zbl

[15] Sloane online encyclopedia of integer sequences, http://oeis.org/wiki/Sum_of_divisors_function

[16] Caratheodory C., Theory of functions of a complex variable, v. 2, Chalsy Publ. Comp., New York, 1954 | MR

[17] Chakravarty S., Ablowitz M. J., Parameterizations of the Chazy equation, 2009, arXiv: 0902.3468v1 | MR

[18] Zagier D., “Elliptic modular forms and their applications”, The 1–2-3 on modular forms, eds. Bruinier J. H. et al., Springer, Berlin, 2008 | MR

[19] Abramowitz M., Stegun I., Handbook of mathematical functions, Dover, New York, 1972 | MR

[20] Ramanujan S., “On certain arithmetical functions”, Trans. Camb. Philos. Soc., 22 (1916), 159–184; Hardy G. H. et al. (eds.), Collected papers of Srinivasa Ramanujan, C.U.P., Cambridge, 1927 | MR

[21] Toh P. O., “Differential equations satisfied by Eisenstein series of level 2”, Ramanujan J., 25 (2011), 179–194 | DOI | MR | Zbl

[22] Huard J. G. et al., “Elementary evaluation of certain convolution sums involving divisor functions”, Number theory for the millennium II, eds. Bennett M. A. et al., A. K. Peters, Natick, Massachusetts, 2002, 229–274 | MR | Zbl

[23] Lagarias J. C., “An elementary problem equivalent to the Riemann hypothesis”, Mathematical Monthly, 109:6 (2002), 534–543 | DOI | MR | Zbl

[24] Varin V. P., “Ploskie razlozheniya i ikh prilozheniya”, Zh. vychisl. matem. i matem. fiz., 55:5 (2015), 807–821 | DOI | Zbl

[25] Nesterenko Y. V., Philippon P. (eds.), “Algebraic independence for values of Ramanujan functions”, Introduction to algebraic independence theory, Springer, Berlin, 2001 | DOI | MR | Zbl

[26] Gradshteyn I. S., Ryzhik I. M., Table of integrals, series, and products, 7th ed., Academic Press, Elsevier, 2007 | MR | Zbl

[27] Whittaker E. T., Watson G. N., A course of modern analysis, 4th ed., C.U.P., Cambridge, 1927 | MR | Zbl

[28] Jacobi C. G. J., Fundamenta nova theoriae functionum ellipticarum, Jacobi's Gesammelte Werke, Chelsea, New York, 1969 | MR

[29] Robin G., “Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann”, J. de Math. Pures et Appliquees, Neuvieme Serie, 63:2 (1984), 187–213 | MR | Zbl