On the characterization of rational difference equations

We present some new results regarding the boundedness character of the kth order rational difference equation; n [ N:When applied to the general fourth order rational difference equation, these results prove the existence of unbounded solutions for 49 special cases of the fourth order rational difference equation, where the boundedness character has not been established yet. This resolves 49 conjectures posed by E. Camouzis and G. Ladas.

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“…[9] when referring to the kth order rational difference equation. Similarly, we shall make extensive use of this notation.…”

Section: Some General Unboundedness Resultsmentioning

confidence: 99%

Unboundedness results for fourth order rational difference equations

Lugo

Palladino

2012

Journal of Difference Equations and Applications

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“…To be more specific if this occurs, then there is a nontrivial subspace of initial conditions where the solution behaves linearly. In [19] and [20], the author shows that the rational difference equation inheirits trichotomy behavior from the associated linear difference equation in this case. To give a demonstration of this idea consider the most basic case, namely the rational difference equation where there is a single delay present in the numerator and every multiple of that delay is not present in the denominator.…”

Section: A Family Of Periodic Trichotomiesmentioning

confidence: 92%

“…To get around this difficulty we must assume that the matrix, which describes the behavior on the invariant subspace where our equation acts linearly, is Hermitian. Under this assumption monotonicity is replaced by monotonicity in norm, at which point theorems 1 and 2 from [19] are applied. Using this approach the proof goes through in many cases.…”

Section: A Family Of Periodic Trichotomiesmentioning

confidence: 99%

On systems of rational difference equations and periodic tetrachotomies

Palladino¹

2009

Preprint

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“…These sets are used extensively in [Palladino 2009b] when referring to the k-th order rational difference equation. Similarly we shall make extensive use of this notation.…”

Section: Congruences For Han's Generating Functionmentioning

confidence: 99%

Untitled

2009

Involve

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Generating and zeta functions, structure, spectral and analytic properties of the moments of the Minkowski question mark function Giedrius Alkauskas (Communicated by Ken Ono) In this paper we are interested in moments of the Minkowski question mark function ?(x). It appears that, to some extent, the results are analogous to results obtained for objects associated with Maass wave forms: period functions, Lseries, distributions. These objects can be naturally defined for ?(x) as well. Various previous investigations of ?(x) are mainly motivated from the perspective of metric number theory, Hausdorff dimension, singularity and generalizations. In this work it is shown that analytic and spectral properties of various integral transforms of ?(x) do reveal significant information about the question mark function. We prove asymptotic and structural results about the moments, calculate certain integrals which involve ?(x), define an associated zeta function, generating functions, Fourier series, and establish intrinsic relations among these objects.

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On the characterization of rational difference equations

Cited by 11 publications

References 6 publications

Unboundedness results for fourth order rational difference equations

Unboundedness results for fourth order rational difference equations

On systems of rational difference equations and periodic tetrachotomies

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