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Applied mathematics (6)

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Fibonacci Quarterly (6)

Authors01f9023f-d29d-4a5e-9798-89f4c5851ab9 (1)81a26182-8169-4390-b85f-b5fad16813cb (1)a90cc4fe-8caf-4587-ab1d-ca6360cecf5b (1)afaa2e14-da18-425a-9673-04ed98ca29b9 (1)b2cc204a-003d-4451-9a59-f5f713de522f (1)View MoreYear (Issue Date)2016-08 (2)2014-02 (1)2015-05 (1)2015-08 (1)2015-11 (1)TypesArticle (6)
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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.

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.

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 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.

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.

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