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)=a₀+a₁f₁ⁿ+⋯+aₗfₗⁿ and G(n)=b₀+b₁g₁ⁿ+⋯+bₘgₘⁿ, has been solved, where the fᵢ and gⱼ are nonconstant entire functions and aᵢ and bⱼ are non-zero constants except that a₀ can be zero. In this paper, we consider the generalization of this problem in which we allow aᵢ and bⱼ to be small growth entire functions with respect to (g₁, ⋯, gₘ) 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.