Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 046, 53 pages      arXiv:1809.00122

Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin

Alexander V. Kitaev
Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia

Received November 13, 2018, in final form May 30, 2019; Published online June 18, 2019

We prove that there exists the unique odd meromorphic solution of dP3, $u(\tau)$ such that $u(0)=0$, and study some of its properties, mainly: the coefficients of its Taylor expansion at the origin andasymptotic behaviour as $\tau\to+\infty$.

Key words: Painlevé equation; asymptotic expansion; hypergeometric function; isomonodromy deformation; greatest common divisor.

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