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Discrete mathematics (23)

Applied mathematics (18)Horadam sequence (1)Obituary (1)Proceedings (1)View MoreJournalBulletin of the Institute of Combinatorics and its Applications (ICA) (9)Fibonacci Quarterly (9)Utilitas Mathematica (3)Carpathian Journal of Mathematics (1)Electronic Notes in Discrete Mathematics (1)AuthorsBagdasar, Ovidiu (3)01f9023f-d29d-4a5e-9798-89f4c5851ab9 (1)3851d3cb-1aae-4e45-a847-2b37647b2398 (1)3fb3b7f2-5bf9-4c67-8512-eb94b785c65b (1)65cd2cf5-e2a3-442a-9d7b-fa73f3f0a44b (1)View MoreYear (Issue Date)2014-05 (4)2016-08 (2)2016-09 (2)2014 (1)2014-02 (1)View MoreTypesArticle (23)
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The Asymptotic Form of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$: Two Proofs

Larcombe, Peter J.; Kirschenhofer, Peter; Fennessey, Eric J. (The Institute of Combinatorics and its Applications, 2014)

On a result of Bunder involving horadam sequences: A new proof

Larcombe, Peter J.; Bagdasar, Ovidiu; Fennessey, Eric J. (Fibonacci Association, 2014-05-02)

This note offers a new proof of a 1975 result due to M. W. Bunder which has recently been proven (inductively), extended empirically and generalized in this journal. The proof methodology, while interesting, cannot be applied realistically beyond the original order two case of Bunder dealt with here.

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.

Closed form evaluations of some series involving Catalan numbers

Larcombe, Peter J. (The Institute of Combinatorics and its Applications (ICA), 2014-05)

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.

On cyclicity and density of some Catalan polynomial sequences

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

We give proofs of cyclic and density properties of some sequences generated by Catalan polynomials, and other observations.

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.

Some factorisation and divisibility properties of Catalan polynomials

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

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 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.

Alwyn Francis Horadam, 1923-2016: A personal tribute to the man and his sequence

Larcombe, Peter J. (The Institute of Combinatorics and its Applications, 2016-09)

Having received news of the passing of Alwyn Horadam this last July, I was determined that I should write something in his honour in which my own contact with him is described and combined with some introductory details on what I feel is his major endowment to the community of mathematicians - the so called and pre-eminent Horadam sequence whose specialisations thereof are great in number.

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