A note on value distribution of solutions of certain second order ODEs

Abstract

While value distribution of solutions of Painlevé equations P 1 − P 6 P_1-P_6 has been thoroughly examined, so-called Painlevé sigma equations S 1 , S 2 S_1, S_2 and S 4 S_4 are relatively less recognized. Various properties of S 1 , S 2 S_1,\,S_2 and S 4 S_4 can be derived both from the form of the equations themselves and from their connection through Hamiltonian systems with P 1 , P 2 P_1,\,P_2 and P 4 P_4 , respectively. This includes distribution of zeros, asymptotic form of the second main theorem and behavior with respect to small target functions. Apart from usual Nevanlinna functionals, also Petrenko’s function of deviation plays a role in the estimates. Additionally, results concerning distribution of values are presented for the modified third Painlevé equation in the final part of the paper.