Meromorphic solutions of P_4,34 and their value distribution

Abstract

Abstract. The unified equation P 4,34 is closely related to the well-known Painlevé equations P 2 and P 4 . We discuss various properties of solutions of P 4,34 , 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 P 4,34 from the perspective of Petrenko's theory, which is also new for P 2 , P 4 and P 34 . We give estimates of deviations and analyse the sets of exceptional values in the sense of Petrenko for equations P 2 , P 4 , P 34 and the unified equation P 4,34 .