Katsuya Ishizaki

Wiman-Valiron method for difference equations

Let f(z) be an entire function of order less than 1/2. We consider an analogue of the Wiman-Valiron theory rewriting power series of f(z) into binomial series. As an application, it is shown that if a transcendental entire solution f(z) of a linear difference equation is of order χ 1/2, then we have log M (r, f) = Lr χ (1 + o(1)) with a constant L > 0.

A Note on the Functional Equation f n + g n + h n = 1 and Some Complex Differential Equations

We consider entire and meromorphic solutions of the functional equation fn + gn + hn = 1. We give new proofs for the known results about the non-existence of transcendental meromorphic solutions for n ≥ 9 and the non-existence of transcendental entire solutions if n ≥ 7. It is shown that if there exist transcendental meromorphic functions f, g and h satisfying the functional equation f8 + g8 + h8 = 1, then f, g and h satisfy the differential equation W(f8, g8, h8) = a(z)(f(z)g(z)h(z))6, where a(z) is a small function with respect to f, g and h.

Meromorphic functions sharing small functions

Abstract. Let f and g be meromorphic functions sharing four small functions

Deficiency for meromorphic solutions of schröder equations

a_1, a_2, a_3, a_4

Non-linear differential equations with transcendental meromorphic solutions

ignoring multiplicities. If there is a small function

Meromorphic solutions of functional equations f(G(z))=R(f(z))

a_5