Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 617-638
University of Szczecin, Institute of Mathematics
ul. Wielkopolska 15, 70-451 Szczecin, Poland; ewa.ciechanowicz 'at' usz.edu.pl
University of Warsaw, Department of Mathematics, Informatics and Mechanics
Banacha 2, 02-097, Warsaw, Poland; filipuk 'at' mimuw.edu.pl
Abstract. The unified equation P4,34 is closely related to the well-known Painlevé equations P2 and P4. We discuss various properties of solutions of P4,34, including one-parameter families of solutions, Bäcklund transformations, regular systems for expansions around zeros and poles and value distribution. In particular, we give estimates of defects and multiplicity indices of transcendental meromorphic solutions of this equation. Moreover, we study solutions of P4,34 from the perspective of Petrenko's theory, which is also new for P2, P4 and P34. We give estimates of deviations and analyse the sets of exceptional values in the sense of Petrenko for equations P2, P4, P34 and the unified equation P4,34.
2010 Mathematics Subject Classification: Primary 30D35; Secondary 34M05, 34M55.
Key words: Meromorphic function, Painlevé equation, defect, deviation, ramification index.
Reference to this article: E. Ciechanowicz and G. Filipuk: Meromorphic solutions of P4,34 and their value distribution. Ann. Acad. Sci. Fenn. Math. 41 (2016), 617-638.
doi:10.5186/aasfm.2016.4146
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