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SubjectsDiscrete mathematics (9)Applied mathematics (6)Horadam sequences (3)Recurrent sequences (3)Periodic sequences (2)View MoreJournal

Fibonacci Quarterly (15)

Authors0000-0003-4193-9842 (5)01f9023f-d29d-4a5e-9798-89f4c5851ab9 (1)25e57dda-3bfa-48ca-a47f-b094c1d90868 (1)3851d3cb-1aae-4e45-a847-2b37647b2398 (1)3fb3b7f2-5bf9-4c67-8512-eb94b785c65b (1)View MoreYear (Issue Date)2016-08 (2)2013-02-01 (1)2013-05-03 (1)2013-11-01 (1)2014-02 (1)View MoreTypesArticle (15)
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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 note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs

Larcombe, Peter J. (The Fibonacci Association, 2015-11)

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.

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.

On a result of Bunder involving Horadam sequences: A proof and generalization

Larcombe, Peter J.; Bagdasar, Ovidiu (Fibonacci Association, 2013-05-03)

This note introduces, proves, extends empirically and generalizes a short 1975 offering of M.W. Bunder who, in this journal, gave an isolated observation involving Horadam sequences on which work has been conducted for nearly half a century.

A generating function approach to the automated evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations.

Larcombe, Peter J.; O'Neill, Sam T. (The Fibonacci Association, 2018-05)

Based on a previous technique deployed in some speciﬁc low order cases, we develop an automated computational procedure to evaluate instances within a class of inﬁnite series comprising exponentiated multiples of generalized linear combinations of Catalan numbers. The methodology is explained, and new results given.

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.

On the number of complex horadam sequences with a fixed period

Bagdasar, Ovidiu; Larcombe, Peter J. (Fibonacci Association, 2013-11-01)

The Horadam sequence is a direct generalization of the Fibonacci numbers in the complex plane, depending on a family of four complex parameters: two recurrence coefficients and two initial conditions. Here the Horadam sequences with a given period are enumerated. The result generates a new integer sequence whose representation involves some well-known functions such as Euler's totient function φ and the number of divisors function ω.

A non-linear identity for a particular class of polynomial families

Larcombe, Peter J.; Fennessey, Eric J. (The Fibonacci Association, 2014-02)

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 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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