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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
(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 cyclicity and density of some Catalan polynomial sequences
(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 new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power
(The Institute of Combinatorics and its Applications (ICA), 2016-01)
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.
Generalised Catalan polynomials and their properties
(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
(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.
On the evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations using a hypergeometric approach
(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.
The Asymptotic Form of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$: Two Proofs
(The Institute of Combinatorics and its Applications, 2014)
On the phenomenon of masked periodic Horadam sequences
(The Institute of Combinatorics and its Applications (ICA), 2015)
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.
A note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs
(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 horadam sequence periodicity: A new approach
(The Institute of Combinatorics and its Applications (ICA), 2015-01)
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.