Special solutions to Chazy equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 2, pp. 210-236
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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.
@article{ZVMMF_2017_57_2_a1,
author = {V. P. Varin},
title = {Special solutions to {Chazy} equation},
journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
pages = {210--236},
publisher = {mathdoc},
volume = {57},
number = {2},
year = {2017},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_2_a1/}
}
V. P. Varin. Special solutions to Chazy equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 2, pp. 210-236. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_2_a1/