Permutability of entire functions satisfying certain differential equations (Q1099289)

Two entire functions f(z) and g(z) are said to be permutable if they satisfy the relation \[ f(g(z))=g(f(z)) \] in the z-plane. In this paper, we have investigated the permutability of entire functions combining the differential equations, i.e. under what condition, two entire functions are permutable, and obtained the result that under the conditions \(0<\lambda (f)\leq \rho (f)<+\infty\) and \(\rho (g)<+\infty\) (\(\lambda\) (f) and \(\rho\) (f) denote the lower order and order of f(z)), if f(z) satisfies a linear differential equation with coefficients of polynomials, then so does g(z). Then with the help of the above result, we completely determined the entire functions permutable with the entire functions like the two forms: \[ f(z)=Q+He\quad P\quad and\quad f(z)=\sin P, \] where Q, H (\(\not\equiv 0)\) and P (\(\not\equiv\) constant) are all polynomials. The former form is a generalization of \textit{T. Kobayashi}'s theorem 1 [see Kodai Math. J. 3, 8-25 (1980; Zbl 0433.30023)].