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...

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...

This paper concerns the oscillation of solutions to the second sublinear dynamic equation with damping xΔΔ t q t xΔ σ t p t x σ t 0, on an isolated time scale which is unbounded above. In 0 < α < 1, α is the quotient of odd positive integers. As an application, we get the difference equation Δ2x n n−γΔx n 1 1/n lnn β b −1 / lnn β x n 1 0, where γ > 0, β > 0, and b is any real number, is oscilla...

and then showing that bn/an converges to ζ(3) fast enough to ensure irrationality of ζ(3) (see [5]). Another remarkable discovery of Apéry is that an and bn satisfy the recursive relation (n+2)un+2− (34n+153n+231n+117)un+1+(n+1)un = 0 (un = an or bn). Thus, if we set A(t) = ∑∞ n=0 ant n and B(t) = ∑∞ n=0 bnt , then the functions A(t) and B(t) satisfy the differential equations (1) (1− 34t+ t)θA...

Difference equations which may arise as discrete approximations to two-point boundary value problems for systems of second-order, ordinary differential equations are investigated and conditions are formulated under which solutions to the discrete problem are unique. Some existence, uniqueness implies existence, and convergence theorems for solutions to the discrete problem are also presented. c...