Let {φn} be a sequence of rational functions with arbitrary complex poles, generated by a certain three-term recurrence relation. In this paper we show that under some mild conditions, the rational functions φn form an orthonormal system with respect to a Hermitian positive-definite inner product.

In 1.1 , the given real sequences {pn}, {qn} satisfy pn T pn > 0, qn T qn for any n ∈ Z, f : Z×R → R is continuous in the second variable, and f n T, z f n, z for a given positive integer T and for all n, z ∈ Z×R. −1 δ −1, δ > 0, and δ is the ratio of odd positive integers. By a solution of 1.1 , we mean a real sequence x {xn}, n ∈ Z, satisfying 1.1 . In 1, 2 , the qualitative behavior of linea...

Keywords: Systems of functional-difference equations Bounded continuous solutions on half-intervals a b s t r a c t Some sufficient conditions for the existence of families of bounded continuous solutions of some classes of systems of functional-difference equations on the intervals ½0; þ1Þ and ðÀ1; 0Š are given.

We show that the p-periodic logistic equation xn+1 = μn mod pxn(1 − xn) has cycles (periodic solutions) of minimal periods 1, p, 2p, 3p, .... Then we extend Singer’s theorem to periodic difference equations, and use it to show the p-periodic logistic equation has at most p stable cycles. Also, we present computational methods investigating the stable cycles in case p = 2 and 3.

For the equation with a distributed delay

Table of contents §9.0. Foreword. §9.1. My more general auxiliary functions in the case k = 1, 2, 3. §9.2. Some relations for the functions, considered in §9.1 in the case l = 0. §9.3. Passing to the system of difference equations for the functions, considered in §9.2. §9.4. Passing from (τ, μ), to (α, ν) in the case l = 0. §9.0. Foreword. Let |z| ≥ 1,−3π/2 < arg(z) ≤ π/2, log(z) = ln(|z|) + i ...