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

For n2 we consider a differential operatorL y] z ⁿ y ⁽ⁿ⁾ +P ₁(z)z ^n–1 y ^(n–1) +P ₂ (z)z ^n–2 y ^n–2 + ...+P _n (z)y = y, p ₁ (z), ..., P _n (z) 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 =o, 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 r, 0 < r <- R; and we establish some additional properties of the sequence ₀ (z), ₁ (z),..., _d–1 (z), f _d (z), f _d+1 (z),... Translated from Matematicheskie Zametki, Vol. 20, No. 6, pp. 869–878, December, 1976.