Deficiency for meromorphic solutions of schröder equations

Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations

“…Eremenko and Sodin [8] used methods from complex dynamics to show that the Valiron and Nevanlinna deficient values of meromorphic solutions of the autonomous Schröder equation (1) always coincide with the exceptional values of R(z). Ishizaki and Yanagihara [18] constructed an example showing that this is not true in general for the non-autonomous Schröder equation. They have also applied Nevanlinna theory to study Borel and Julia directions of meromorphic solutions of the Schröder equation [19,20].…”

Growth and poles of meromorphic solutions of some difference equations

“…Valiron has shown that the non-autonomous Schröder q-difference equation f (qz) = R z, f (z) , (1.1) where R(z, f (z)) is rational in both arguments, admits meromorphic solutions if q ∈ C is suitably chosen [13]. The property of meromorphic solutions of (1.1) is deeply investigated during the last decades, see for instance [3,7,10,11]. In 1998, Bergweiler et al [2] pointed out that transcendental meromorphic solutions f (z) of the functional equation n j=0 a j (z) f c j z = Q (z), (1.2) where 0 < |c| < 1 is a complex number, a j (z), j = 0, 1, 2, .…”

Some Results on the Deficiencies of Some Differential-Difference Polynomials of Meromorphic Function

“…Recently, many scholars have exhibited an increasing interest in studying some properties of meromorphic function and its difference by relying on the Nevanlinna theory, and they have produced a number of papers focusing on the value distribution and uniqueness of complex difference, differential-difference operators, differential-difference equations, and so on (see [5][6][7][8][9][10][11]).…”