HAHN'S PROBLEM WITH RESPECT TO SOME PERTURBATIONS OF THE RAISING OPERATOR \(X-c\)

In this paper, we study the Hahn's problem with respect to some raising operators perturbed of the operator \(X-c\), where \(c\) is an arbitrary complex number. More precisely, the two following characterizations hold: up to a normalization, the \(q\)-Hermite (resp. Charlier) polynomial is the only \(H_{\alpha,q}\)-classical (resp. \(\mathcal{S}_{\lambda}\)-classical) orthogonal polynomial, where \(H_{\alpha, q}:=X+\alpha H_q\) and \(\mathcal{S}_{\lambda}:=(X+1)-\lambda\tau_{-1}.\)

On the growth of solutions to second order differential equations in Banach spaces

SynopsisWe obtain estimates for the exponential growth of the solutions to u″(t) = (A + ζ2I)u(t) in terms of the exponential growth of the solutions to u″(t) = Au(t), where ζ is an arbitrary complex number. Estimates in exponentially weighted L2 norms are also considered in Hilbert space.

Applications of the theory of cluster sets to a class of meromorphic functions

Let f (z) be meromorphic and non-rational in the domain |z| < R ≤ ∞, and let a be an arbitrary complex number, which may be infinite. The deficiency δ(a) of the value a is defined bywhere m(r, a), N(r, a) and T(r) are defined as usual (cf. (10), pp. 156 ff.). For the class of functions considered in this paper the characteristic function T(r) is unbounded, and this will be assumed throughout. The upper (or Valiron) deficiency (16) of the value a is denned byfrom which it follows that 0 ≤ δ(a) ≤ Δ(a) ≤ 1. A value a for which Δ(a) > 0 is called exceptional or deficient, and a value for which Δ(a) = 0 is called normal. We shall denote by G[a, σ] the open set of all values z in | z | < R for which | f(z) – a | < σ, where σ is a given positive number; we shall say that a component Gn[a, σ] of G[a, a] is bounded if the closure G¯n[a, σ] is contained in | z | < R, otherwise Gn[a, σ] will be called unbounded. In the case a = ∞, it is natural to define Gn[∞, σ] as the set of all z for which | f(z) | > 1/σ.