Quotient Problem For Entire Functions with Moving Targets

As an analogue of the Hadamard quotient problem in number theory, the quotient problem (in the sense of complex entire functions) for two sequences $F(n)=a_0+a_1f_1^n+\cdots+a_lf_l^n$ and $G(n)=b_0+b_1g_1^n+\cdots+b_mg_m^n$, has been solved, where the $f_i$ and $g_j$ are nonconstant entire functions and $a_i$ and $b_j$ are non-zero constants except that $a_0$ can be zero. In this paper, we consider the generalization of this problem in which we allow $a_i$ and $b_j$ to be small growth entire functions with respect to $(g_1, \cdots, g_m)$ by modifying the second main theorem with moving targets to a truncated version. We also compare our result to a special case in exponential polynomials first studied by Ritt.