# On sequence-based closed form entries for an exponentiated general $2 \times 2$ matrix: A re-formulation and an application.

Journal article

Larcombe, Peter J. and Fennessey, Eric J. 2017. On sequence-based closed form entries for an exponentiated general $2 \times 2$ matrix: A re-formulation and an application.

*Bulletin of the Institute of Combinatorics and its Applications (ICA).*Authors | Larcombe, Peter J. and Fennessey, Eric J. |
---|---|

Abstract | Closed form entries for an exponentiated (and arbitrary) 2 ⇥ 2 matrix are established here, and expressed in terms of a specialized Horadam sequence; two proofs of the result are given accordingly, along with examples and observations derived therefrom. The result o↵ers a new formulation of a general class of polynomial families associated with sequences whose ordinary generating functions are governed by quadratic equations. |

Keywords | Sequences; Matrices; Horadam sequence |

Year | 2017 |

Journal | Bulletin of the Institute of Combinatorics and its Applications (ICA) |

Publisher | The Institute of Combinatorics and its Applications |

ISSN | 11831278 |

Web address (URL) | http://hdl.handle.net/10545/622398 |

hdl:10545/622398 | |

Publication dates | Jan 2017 |

Publication process dates | |

Deposited | 21 Mar 2018, 10:26 |

Contributors | University of Derby |

File | File Access Level Open |

File | File Access Level Open |

Permalink -

https://repository.derby.ac.uk/item/92746/on-sequence-based-closed-form-entries-for-an-exponentiated-general-2-times-2-matrix-a-re-formulation-and-an-application

## Download files

##### 30

total views##### 17

total downloads##### 0

views this month##### 0

downloads this month

## Export as

## Related outputs

##### Reflections On What Mathematics Is and Isn't: Halmos, Keyser, and Others

Larcombe, P. 2022. Reflections On What Mathematics Is and Isn't: Halmos, Keyser, and Others.*Palestine Journal of Mathematics.*11 (3), pp. 664-699.

##### Mathematician? Feeling Old? A Discussion

Larcombe, P. 2022. Mathematician? Feeling Old? A Discussion.*Palestine Journal of Mathematics.*11, pp. 19-36.

##### On 'Two Cultures' and Tackling the 'Writing Versus Mathematics' Dichotomy

Larcombe, P. 2021. On 'Two Cultures' and Tackling the 'Writing Versus Mathematics' Dichotomy.*Palestine Journal of Mathematics.*10 (2), pp. 480-486.

##### Write, and Write Well - Speak, and Speak Well: The Gospel According to Halmos and Rota

Larcombe, P. 2021. Write, and Write Well - Speak, and Speak Well: The Gospel According to Halmos and Rota.*Palestine Journal of Mathematics.*10 (1), pp. 86-101.

##### Mathematicians can also write, right?

Larcombe, Peter J 2019. Mathematicians can also write, right?*Mathematics Today.*

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

Larcombe, Peter J. and O'Neill, Sam T. 2018. A generating function approach to the automated evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations.*Fibonacci Quarterly.*

##### New proofs of linear recurrence identities for terms of the Horadam sequence.

Larcombe, Peter J. and Fennessey, Eric J. 2019. New proofs of linear recurrence identities for terms of the Horadam sequence.*Palestine Journal of Mathematics.*

##### On the notion of mathematical genius: rhetoric and reality.

Larcombe, Peter J. 2019. On the notion of mathematical genius: rhetoric and reality.*Palestine Journal of Mathematics.*

##### On generalised multi-index non-linear recursion identities for terms of the Horadam sequence.

Larcombe, Peter J. and Fennessey, Eric J. 2019. On generalised multi-index non-linear recursion identities for terms of the Horadam sequence.*Palestine Journal of Mathematics.*

##### A note on two rational invariants for a particular 2 x 2 matrix.

Larcombe, Peter J. and Fennessey, Eric J. 2018. A note on two rational invariants for a particular 2 x 2 matrix.*Palestine Journal of Mathematics.*

##### A new non-linear recurrence identity class for Horadam sequence terms.

Larcombe, Peter J. and Fennessey, Eric J. 2018. A new non-linear recurrence identity class for Horadam sequence terms.*Palestine Journal of Mathematics.*

##### On two derivative sequences from scaled geometric mean sequence terms.

Larcombe, Peter J., Rabago, Julius, F. T. and Fennessey, Eric J. 2018. On two derivative sequences from scaled geometric mean sequence terms.*Palestine Journal of Mathematics.*

##### A new tri-diagonal matrix invariance property.

Larcombe, Peter J. and Fennessey, Eric J. 2017. A new tri-diagonal matrix invariance property.*Palestine Journal of Mathematics.*

##### A few thoughts on the aesthetics of mathematics in research and teaching.

Larcombe, Peter J. 2018. A few thoughts on the aesthetics of mathematics in research and teaching.*Palestine Journal of Mathematics.*

##### A closed form formulation for the general term of a scaled triple power product recurrence sequence.

Larcombe, Peter J. and Fennessey, Eric J. 2017. A closed form formulation for the general term of a scaled triple power product recurrence sequence.*Fibonacci Quarterly.*

##### Opinions, opinions, opinions ...

Larcombe, Peter J. 2017. Opinions, opinions, opinions ...*Mathematics Today.*

##### Mathematics as a mirror of painting.

Larcombe, Peter J. 2017. Mathematics as a mirror of painting.*Mathematics Today.*

##### Horadam sequences: A survey update and extension.

Larcombe, Peter J. 2017. Horadam sequences: A survey update and extension.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### ON THE MASKED PERIODICITY OF HORADAM SEQUENCES: A GENERATOR-BASED APPROACH

Bagdasar, Ovidiu D., Larcombe, Peter J. and Bagdasar, O. 2017. ON THE MASKED PERIODICITY OF HORADAM SEQUENCES: A GENERATOR-BASED APPROACH.*Fibonacci Quarterly.*

##### Conditions governing cross-family member equality in a particular class of polynomial families

Larcombe, Peter J. and Fennessey, Eric J. 2014. Conditions governing cross-family member equality in a particular class of polynomial families.*Fibonacci Quarterly.*

##### On certain series expansions of the sine function: Catalan numbers and convergence

Larcombe, Peter J., O'Neill, Sam T. and Fennessey, Eric J. 2014. On certain series expansions of the sine function: Catalan numbers and convergence.*Fibonacci Quarterly.*

##### On the number of complex horadam sequences with a fixed period

Bagdasar, Ovidiu and Larcombe, Peter J. 2013. On the number of complex horadam sequences with a fixed period.*Fibonacci Quarterly.*

##### On the characterization of periodic complex Horadam sequences

Bagdasar, Ovidiu and Larcombe, Peter J. 2013. On the characterization of periodic complex Horadam sequences.*Fibonacci Quarterly.*

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

Larcombe, Peter J. and Bagdasar, Ovidiu 2013. On a result of Bunder involving Horadam sequences: A proof and generalization.*Fibonacci Quarterly.*

##### On the characterization of periodic generalized Horadam sequences

Bagdasar, Ovidiu and Larcombe, Peter J. 2014. On the characterization of periodic generalized Horadam sequences.*Journal of Difference Equations and Applications.*https://doi.org/10.1080/10236198.2014.891022

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

Larcombe, Peter J. 2016. Alwyn Francis Horadam, 1923-2016: A personal tribute to the man and his sequence.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### A scaled power product recurrence examined using matrix methods

Larcombe, Peter J. and Fennessey, Eric J. 2016. A scaled power product recurrence examined using matrix methods.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

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

Larcombe, Peter J. 2016. On the evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations using a hypergeometric approach.*Fibonacci Quarterly.*

##### On a scaled balanced-power product recurrence

Larcombe, Peter J. and Fennessey, Eric J. 2016. On a scaled balanced-power product recurrence.*Fibonacci Quarterly.*

##### A short monograph on exposition and the emotive nature of research and publishing

Larcombe, Peter J. 2016. A short monograph on exposition and the emotive nature of research and publishing.*Mathematics Today.*

##### A polynomial based construction of periodic Horadam sequences

Larcombe, Peter J. and Fennessey, Eric J. 2016. A polynomial based construction of periodic Horadam sequences.*Utilitas Mathematica.*

##### A new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power

Larcombe, Peter J. 2016. A new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### On the jacobsthal, horadam and geometric mean sequences

Larcombe, Peter J. and Rabago, Julius, F. T. 2016. On the jacobsthal, horadam and geometric mean sequences.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### A note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs

Larcombe, Peter J. 2015. A note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs.*Fibonacci Quarterly.*

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

Larcombe, Peter J. 2015. Closed form evaluations of some series comprising sums of exponentiated multiples of two-term and three-term Catalan number linear combinations.*Fibonacci Quarterly.*

##### Closed form evaluations of some series involving Catalan numbers

Larcombe, Peter J. 2014. Closed form evaluations of some series involving Catalan numbers.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### 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. and Fennessey, Eric J. 2015. A condition for anti-diagonals product invariance across powers of $2 \times 2$ matrix sets characterizing a particular class of polynomial families.*Fibonacci Quarterly.*

##### On the phenomenon of masked periodic Horadam sequences

Larcombe, Peter J. and Fennessey, Eric J. 2015. On the phenomenon of masked periodic Horadam sequences.*Utilitas Mathematica.*

##### On horadam sequence periodicity: A new approach

Larcombe, Peter J. and Fennessey, Eric J. 2015. On horadam sequence periodicity: A new approach.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

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

Larcombe, Peter J. and Fennessey, Eric J. 2014. A non-linear identity for a particular class of polynomial families.*Fibonacci Quarterly.*

##### On cyclicity and density of some Catalan polynomial sequences

Larcombe, Peter J. and Fennessey, Eric J. 2014. On cyclicity and density of some Catalan polynomial sequences.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### Some factorisation and divisibility properties of Catalan polynomials

Jarvis, Frazer A., Larcombe, Peter J. and Fennessey, Eric J. 2014. Some factorisation and divisibility properties of Catalan polynomials.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### Generalised Catalan polynomials and their properties

Larcombe, Peter J., Jarvis, Frazer A. and Fennessey, Eric J. 2014. Generalised Catalan polynomials and their properties.*Bulletin of the Institute of Combinatorics and its Applications (ICA).*

##### The Asymptotic Form of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$: Two Proofs

Larcombe, Peter J., Kirschenhofer, Peter and Fennessey, Eric J. 2014. The Asymptotic Form of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$: Two Proofs.*Utilitas Mathematica.*

##### On the structure of periodic complex Horadam orbits

Bagdasar, Ovidiu D., Larcombe, Peter J., Anjum, Ashiq and Bagdasar, O. 2016. On the structure of periodic complex Horadam orbits.*Carpathian Journal of Mathematics.*

##### ON A RESULT OF BUNDER INVOLVING HORADAM SEQUENCES: A NEW PROOF

Larcombe, Peter J., Bagdasar, O. and Fennessey, Eric J. 2014. ON A RESULT OF BUNDER INVOLVING HORADAM SEQUENCES: A NEW PROOF.*Fibonacci Quarterly.*pp. 175-177.