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Applied mathematics (18)

Discrete mathematics (18)Proceedings (1)Pure mathematics (1)View MoreJournalBulletin of the Institute of Combinatorics and its Applications (ICA) (7)Fibonacci Quarterly (6)Utilitas Mathematica (3)Carpathian Journal of Mathematics (1)Electronic Notes in Discrete Mathematics (1)AuthorsLarcombe, Peter J. (17)Fennessey, Eric J. (10)Bagdasar, Ovidiu (2)Bagdasar, Ovidiu (2) Jarvis, Frazer A. (2)View MoreYear (Issue Date)2014-05 (4)2016-08 (2)2014 (1)2014-02 (1)2015 (1)View MoreTypesArticle (18)
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Generalised Catalan polynomials and their properties

Larcombe, Peter J.; Jarvis, Frazer A.; Fennessey, Eric J. (The Institute of Combinatorics and its Applications, 2014-05)

We introduce a new type of polynomial, termed a generalised Catalan polynomial. We list essential mathematical properties and give two associated combinatorial interpretations.

On a scaled balanced-power product recurrence

Larcombe, Peter J.; Fennessey, Eric J. (The Fibonacci Association, 2016-08)

A power product recurrence (due to M. W. Bunder) is extended here by the introduction of a scaling factor, and delivers a sequence whose general term closed form is derived for both degenerate and non-degenerate characteristic root cases. It is shown how recurrence parameter conditions dictate the nature of each solution type, and a fundamental link between them is highlighted together with some other observations and results.

A condition for anti-diagonals product invariance across powers of $2 \times 2$ matrix sets characterizing a particular class of polynomial families

Larcombe, Peter J.; Fennessey, Eric J. (The Fibonacci Association, 2015-05)

Motivated by some recent work on a particular class of polynomial families associated with certain types of integer sequences, we formulate a sufficient condition under which the anti-diagonals products across sets of characterizing 2 × 2 matrices remain invariant as matrix power increases. Two proofs are given along with some examples.

Closed form evaluations of some series comprising sums of exponentiated multiples of two-term and three-term Catalan number linear combinations

Larcombe, Peter J. (The Fibonacci Association, 2015-08)

Closed form evaluations of some infinite series comprising sums of exponentiated multiples of two-term and three-term Catalan number linear combinations are presented using three contrasting approaches. Known power series expansions of the trigonometric functions sin(4α) and sin(6α) each readily give a set of (four) results which are reformulated via a hypergeometric route and, additionally, using only the generating function for the Catalan sequence; the latter two methods are shown to be connected.

A polynomial based construction of periodic Horadam sequences

Larcombe, Peter J.; Fennessey, Eric J. (The Institute of Combinatorics and its Applications (ICA), 2016-03)

A recent matrix based approach to the study of self-repeating Horadam sequences has identified a mechanism to produce guaranteed (and arbitrary) periodicity through a novel formulation for the two governing parameters in the defining linear Horadam recurrence equation. We consider this further here, giving supporting examples to illustrate the methodology which utilises so called Catalan polynomials.

On the structure of periodic complex Horadam orbits

Bagdasar, Ovidiu; Larcombe, Peter J.; Anjum, Ashiq (North University of Baia Mare (Romania), 2016-01-16)

Numerous geometric patterns identified in nature, art or science can be generated from recurrent sequences, such as for example certain fractals or Fermat’s spiral. Fibonacci numbers in particular have been used to design search techniques, pseudo random-number generators and data structures. Complex Horadam sequences are a natural extension of Fibonacci sequence to complex numbers, involving four parameters (two initial values and two in the defining recursion), therefore successive sequence terms can be visualized in the complex plane. Here, a classification of the periodic orbits is proposed, based on divisibility relations between orders of generators (roots of the characteristic polynomial). Regular star polygons, bipartite graphs and multisymmetric patterns can be recovered for selected parameter values. Some applications are also suggested.

On the evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations using a hypergeometric approach

Larcombe, Peter J. (The Fibonacci Association, 2016-08)

Infinite series comprising exponentiated multiples of p-term linear combinations of Catalan numbers arise naturally from a related power series expansion for sin(2pα) (in odd powers of sin(α)) which itself has an interesting history. In this article some explicit results generated previously by the author (for p = 1, 2, 3) are discussed in the context of this general problem of series summation, and new evaluations made for the cases p = 4, 5 by way of further examples. A powerful hypergeometric approach is adopted which offers, from the analytical formulation developed, a means to achieve these particular evaluations and in principle many others for even greater values of p.

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