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On cyclicity and density of some Catalan polynomial sequences
We give proofs of cyclic and density properties of some sequences generated by Catalan polynomials, and other observations.
On the phenomenon of masked periodic Horadam sequences
A recently discovered phenomenon, termed masked periodicity and observed in self-repeating Horadam sequences using matrix based methods in a study by the authors elsewhere, is considered further here. This article continues the approach, identifying both governing parameters and particular behaviour types which fall naturally into three categories convenient for explanation and illustration.
Some factorisation and divisibility properties of Catalan polynomials
We present some factorisation and divisibility properties of Catalan polynomials. Initial results established with ad hoc proofs then make way for a more systematic approach and use of the well developed theory of cyclotomic polynomials.
On horadam sequence periodicity: A new approach
A so called Horadam sequence is one delivered by a general second order recurrence formula with arbitrary initial conditions. We examine aspects of self-repeating Horadam sequences by applying matrix based methods in new ways, and derive some conditions governing their cyclic behaviour. The analysis allows for both real and complex sequence periodicity.
A non-linear identity for a particular class of polynomial families
We state and prove a new non-linear identity for a class of polynomial families associated with integer sequences whose ordinary generating functions have quadratic governing equations with functional (polynomial) coefficients.
On the jacobsthal, horadam and geometric mean sequences
This paper, in considering aspects of the geometric mean sequence, offers new results connecting Jacobsthal and Horadam numbers which are established and then proved independently.
A new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power
We obtain an existing 2004 result of J. McLaughlin which gives explicit entries for a general dimension 2 matrix raised to an arbitrary power. Our formulation employs so called Catalan polynomials related to the crucial parameter of McLaughlin’s statement, and is a new one running along a different line of argument.
A note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs
An invariance matrix property, first observed empirically and seemingly absent from mainstream literature, is stated and established formally here. Four short, and different, proofs are given accordingly.
Closed form evaluations of some series involving Catalan numbers
Closed form evaluations of some infinite series comprising sums of exponentiated multiples of Catalan numbers are yielded by a known expansion of the trigonometric function sin(2α), and then re-formulated independently as verification.