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.