The iterated -order of an entire function is defined as
The finiteness degree of the order of an entire function is defined as if is a polynomial or if is a transcendental entire function and for some , otherwise The iterated convergence exponent of the sequence of distinct zeros of an entire function is defined as
where is the counting function of the distinct zeros of in
In this paper, the author studies the iterated order and iterated convergence exponent of the sequence of distinct zeros of the entire solutions of the following linear differential equations
where and are entire functions. The author proves that if are entire functions with for some , then the equation (1) has at least one solution with and Furthermore, if is an entire function with either or and then there exists at least one solution of the corresponding homogeneous equation (1) of the equation (2) such that all solutions in the solution subspace satisfying and with at most one exception, where is a solution of (2).