We introduce a group theoretical tool on which one can derive a family of identities from sequences that are defined by a recursive relation. As an illustration it is shown that n−1 ∑ i=1 Fn−iF 2 i = 1 2 n ∑ i=1 (−1)(F2i − Fi) = ( Fn+1 2 )

The primary aim of this paper is to (provide tools in order to) compute Galois groups of classical irregular q-difference equations. We are particularly interested in quantizations of omnipresent differential equations in the mathematical and physical literature, namely confluent generalized q-hypergeometric equations and q-Kloosterman equations.

We show that if v is a symmetric regular Laguerre-Hahn linear form (functional), then the linear form u defined by u = −λx−2v + δ0 is also regular and symmetric LaguerreHahn linear form for every complex λ except for a discrete set of numbers depending on v. We explicitly give the coefficients of the second-order recurrence relation, the structure relation of the orthogonal sequence associated ...

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...

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...