On the behavior of solutions of the system of rational difference equations x n+1 = x n−1 y n x n−1 − 1 , y n+1 = y n−1 x n y n−1 − 1 , z n+1 = 1 y n z n Abstract In this article, we investigate the solutions of the system of difference equations y n+1 = y n−1 x n y n−1 − 1 , y n+1 = y n−1 x n y n−1 − 1 , z n+1 = 1 y n z n where x 0 , x-1 , y 0 , y-1 , z 0 , z-1 real numbers such that y 0 x-1 ≠...

We present, for the first time, two hierarchies of nonlinear, integrable q-difference equations, one of which includes a q-difference form of each of the second and fifth Painlevé equations, qPII and qPV, the other includes qPIII. All the equations have multiple free parameters. A method to calculate a 2× 2 Lax pair for each equation in the hierarchy is also given.

We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by explicitly constructing its Hamiltonian formulation and showing that it coincides with that of a Garnier system. Such systems can also be characterised by recurrenc...

This paper studies the essential elements (Puppe, 1996) associated with binary relations over opportunity sets. We restrict attention to binary relations which are reflexive and transitive (pre-orders) and which further satisfy a monotonicity and desirability condition. These are called opportunity relations (ORs). Our main results axiomatically characterise two important classes of ORs: those ...

Asymptotic properties of some non-linear difference equations are considered. AMS(1980) Subject Classification: 39A12, 45E99..

An elevated Schröder path is a lattice path that uses the steps (1, 1), (1,−1), and (2, 0), that begins and ends on the x-axis, and that remains strictly above the x-axis otherwise. The total area of elevated Schröder paths of length 2n + 2 satisfies the recurrence fn+1 = 6fn − fn−1, n ≥ 2, with the initial conditions f0 = 1, f1 = 7. A combinatorial interpretation of this recurrence is given, b...

In this paper, we study the relationship among left peaks, interior peaks and up-down runs of simsun permutations. Properties of the generating polynomials, including the recurrence relation, generating function and real-rootedness are studied. Moreover, we introduce and study simsun permutations of the second kind.

One of the earliest difference equations, the Fibonacci sequence, was introduced in 1202 in “Liberabaci,” a book about the abacus, by the famous Italian Leonardo di Pisa, better known as Fibonacci. The problem may be stated as follows: how many pairs of rabbits will there be after one year when starting with one pair of mature rabbits, if each pair of rabbits give birth to a new pair each month...

In this paper we study the sequences generated by a neuronal equation of the form xn " 1[!ki"1 aixn#i # Θ], where k is the size of the memory. We show that in the case where all the parameters (ai)1%i%k are strictly positive reals and k is a multiple of 6 (i.e., k " 6m), there exists a subset B " &w1, w2, . . . , wΑ( of &1, 2, . . . , m # 1( and for which there exists a neuronal equation of mem...

ArXiv : math.CO/0403017 v 1 1 March 2004 Summary Combinatorial interpretation of the fibonomial coefficients recently attampted by the present author [1,2] and presented here with suitable improvements results in a proposal of a might be combinatorial interpretation of the recurrence relation for fibonomial coefficients . The presentation is provided within the context of the classical combinat...