Eigenfunctions and associated functions of an $n$-th-order linear differential operator

For $n\ge2$ we consider a differential operator
$$ L[y]\equiv z^ny^{(n)}+P_1(z)z^{n-1}y{(n-1)}+_2(z)z^{n-2}Y^{(n-2)}+\dots+P_n(z)y=\mu y;\quad P_1,\dots,P_n(z)\in A_R $$
here $A_R$ is the space of functions which are analytic in the disk $|z|<R$, equipped with the topology of compact convergence. We prove the existence of sequences $\{f_k(z)\}_{k=0}^\infty$, consisting of a finite number of associated functions of the operator $L$ and an infinite number of its eigenfunctions; we show that the sequence forms a basis in $A_r$ for an arbitrary $\{f_k(z)\}_{k=0}^\infty$; and we establish some additional properties of the sequencephiv $\{f_k(z)\}_{k=0}^\infty$