Let [T (n, k)]n,k>0 be a triangle of positive numbers satisfying the three-term recurrence relation T (n, k) = (a1n + a2k + a3)T (n− 1, k) + (b1n + b2k + b3)T (n− 1, k − 1). In this paper, we give a new sufficient condition for linear transformations

In this paper we solve several recurrence relations with two (three) indices using combinatorial methods. Moreover, we present several recurrence relations with two (three) indices related to ternary paths and k-ary paths. © 2007 Elsevier B.V. All rights reserved. MSC: 10A35; 65Q05; 05.10

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.