# Professor Peter J Larcombe

**Position**: Professor of Discrete and Applied Mathematics

**College**: College of Engineering and Technology

**Department**: Computing and Mathematics

**Subject area**: Mathematics

## About

I have recently completed a four year period overseeing the three Mathematics Programmes that we offer at undergraduate level (B.Sc. (Hons.) in Mathematics, Mathematics with Education, and Mathematics and Computer Science), and lead on various outreach activities designed to promote Mathematics as a subject here at the University of Derby.

In 2013 I became the first ever professor of mathematics at the University of Derby [ see the *Derby Telegraph* article at http://www.derbytelegraph.co.uk/discovering-beauty-maths-gaining-key-skills/story-19659407-detail/story.html ], having been appointed reader in 2010.

As well and teaching and pursuing research interests, I undertake the role of Academic Representative at the University of Derby for the I.M.A. (Institute of Mathematics and its Applications).

## Teaching responsibilities

I have delivered and/or led many different modules over the years at Derby, and have recently taught on the following undergraduate modules:

- Linear Algebra (Year 2)
- Mathematical Methods (Year 2)
- Mathematics Group Project (Year 2)
- Non-Linear Systems Dynamics (Year 3)
- Calculus (Year 1)

## Professional interests

I am interested in the mathematical education and welfare of students, and take a keen interest in the wider mathematical field. It is a privilege to be an ambassador for the subject of mathematics through my teaching, outreach and research efforts.

I am very research active in some areas that lie within the field of discrete mathematics, but occasionally I publish on pedagogic and historical matters pertaining to mathematics too.

In addition, I am sometimes moved to reflect and write on certain aspects of the way we mathematicians go about our business, the environments in which we work, and the behaviours that shape (and are simultaneously driven by) each - I have, for instance, published occasional (professional magazine) works covering the role of exposition in mathematics, the emotive nature of research and publishing, and issues surrounding authorship.

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## Research interests

My main research interests cover areas within discrete mathematics which include such things as

- Theory of Integer Sequences
- Hypergeometric Functions (and Identities)
- Iterated Generating Functions
- Binomial Sums (and Identities)
- Asymptotics
- Linear and Non-Linear Recurrence Equations
- Computer Algebra
- Number Theory
- Sequence Polynomial Families
- Periodic Real and Complex Recurrences Sequences

After gaining a B.Sc. Special Honours degree (1st Class) in Mathematics (University of Hull, 1984) and staying on to complete a Ph.D. in Applied Mathematics (modelling ice and snow deposition on overhead electricity power cables; awarded 1988), I moved into the area of control engineering when becoming a postdoctoral researcher in the Control Group of Professor P.J. Gawthrop at the University of Glasgow. My main interests and publications at that time related to the field of control theory and mechanical systems modelling, particularly in regard to the application of computer algebra which was then an emerging branch of mathematical computation receiving ever more interest. Having moved to Derby in 1993 I subsequently developed a strong interest in combinatorics and discrete mathematics (initially brought about through supervision of an undergraduate student project), and have since published a considerable number of works in this area which combine theoretical results with algebraic computation.

My work over the last twenty years or so involves, in essence, the creation/identification and solution of a variety of mathematical problems within combinatorics and discrete mathematics (collaborators include Prof. Dr. W. Koepf (University of Kassel, Germany), Prof. I.M. Gessel (Brandeis University, U.S.A.), Prof. M.E. Larsen (University of Copenhagen, Denmark), Prof. P. Kirschenhofer (University of Leoben, Austria), Dr. R.B. Paris (University of Abertay Dundee)). I naturally undertake some elements of research on an *ad hoc* basis, but my main contribution has been in the area of sequences, evidenced in part by two integer sequences which have been established and formally recognised by name - the so called *Catalan-Larcombe-French* and *Fennessey-Larcombe-French* sequences. Results established by myself, in collaboration with others, have generated considerable interest, and they have been recently examined by Chinese mathematicians looking at their convexity and concavity properties. These two new integer sequences have been formulated from first principles, and they are now named ones within the mathematical community. The sequences are derived from elliptic integrals (of the first and second kind, respectively), which are themselves important mathematical objects that have received much attention in both pure and applied mathematics. The application of a non-linear transformation to these integrals is novel, and allows the emerging sequences to be related since the integrals themselves are connected in a mathematically fundamental way; both sequences are listed on the wonderful *On-Line Encyclopaedia of Integer Sequences* (see Sequence Nos. A053175 & A065409, and the associated entries). I should mention, too, my late colleague David French who passed away in February 2014. David and I worked intensely on the C-L-F and F-L-F sequences (both of which carry his name as a legacy), and on other problems, for about a decade. I owe him a lot for his enthusiasm, dedication and mathematical effort during this period. I still have some of his ideas and analysis to explore, and hope to formulate further results inspired by him even though he is no longer with us.

I also have a particular interest in the Catalan sequence. Since the late 1990s I have, for instance, been examining some unusual power series expansions which involve the celebrated Catalan numbers - this has necessitated looking at, and extending, some work by a Chinese scholar that dates back to the 1600s, and trying to generalise the suite of results resulting therefrom. The Catalan sequence is ubiquitous in mathematics and appears, sometimes rather unexpectedly, in a whole range of counting problems; it is a sequence on which I have published regularly. Its own history is an interesting one, and the 200th anniversary of the birth of Eugene C. Catalan - after whom the sequence is known - was marked by a Special Issue (Vol. 76, May 2014) of the *Bulletin of the I.C.A*. organised by me with invited contributions. The Catalan sequence is certainly the best know sequence, among us mathematicians, after the famous Fibonacci sequence.

Through past work with a completed Ph.D. student (and my great friend Dr. Eric J. Fennessey) I have opened up and begun to explore a new area of discrete mathematics based on the notion of a so called Iterated Generating Function. An I.G.F. - arising from some input/output rule governing general (real or complex) polynomials - is an iterative construct which generates a sequence through the coefficients of its terms as the computations progress. To date we have shown that there exists both finite and infinite sequences for which an I.G.F. algorithm can be formulated, whilst on the other hand there are so called 'impossible' sequences which cannot be realised in this way (this is relevant to the theory of automata). Elsewhere we have seen that when the input/output relationship is a particular instance of a general Householder scheme (which delivers, as separate special cases, the well known Newton-Raphson and Halley root-finding versions prominent in numerical analysis), then its algebraic execution by computer generates a pair of non-linear identities for polynomial families associated with sequences whose generating functions are governed by a quadratic equation; observed initially by empirical computation, we have produced a fully general closed form description of this phenomenon from which pleasing spin-off results are identifiable (for example, any such identity for the family of Schroeder polynomials will yield a new relation, of commensurate degree, for the Delannoy numbers intimately connected to them). These polynomial families themselves have some interesting mathematical aspects that provide further ongoing avenues for research.

Finally, with colleague Dr. O.D. Bagdasar (and Dr. Fennessey), a longstanding construct - a so called Horadam sequence - has been re-examined and new results found. Horadam sequences are second-order linear recurrence sequences which depend on a family of four general parameters (two in the defining recursion itself, and two initial values), being so named after the publication of two seminal papers by Professor Alwyn F. Horadam in the 1960s; the general recursion produces many familiar sequences as particular instances - such as Fibonacci, Pell, Lucas, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas, Tagiuri, Fermat, Fermat-Lucas - and is connected to the famous Chebyshev polynomials of 1st and 2nd kind which can be generated via such a recurrence. While research has continued consistently on a variety of mathematical aspects of Horadam sequences for over half a century now, a new line of enquiry has been followed here at Derby, where we have considered cyclic (that is to say, self-repeating) sequences and identified necessary and sufficient conditions governing periodicity of complex Horadam sequences under general initial values, offering a full characterisation (of degenerate and non-degenerate characteristic solution types) accordingly. We are very excited by this recent development as the phenomenon of periodicity is a new one in the context of Horadam sequences, and the analysis conducted thus far - naturally of mathematical interest *per se* from a theoretical viewpoint - also seems to have potential applications in some practical aspects of computing (namely, in the design of network optimisation, in random number generation and in certain large data searches). The research combines strong mathematical number theory and algebra with algorithmic computation, creating an underpinning theoretical framework to the concept of Horadam cyclicity with some beautiful visual results produced.

## Membership of professional bodies

I hold the following membership of professional bodies:

- Fellow of the Institute of Mathematics and its Applications
- Fellow of the Institute of Combinatorics and its Applications
- Member of the Institution of Engineering and Technology
- Member of the American Mathematical Society
- Member of the Association of Computing Machinery
- Member of S.E.F.I. (European Society for Engineering Education

I am a Chartered Engineer, Chartered Mathematician, and Chartered Scientist.

## Qualifications

### Undergraduate qualifications

- B.Sc. 1st Class (Special Honours) Degree in Mathematics, University of Hull (1984) [ Awarded the 1984 University of Hull Slater Prize in Applied Mathematics ]

### Research qualifications

- Ph.D. in Applied Mathematics, University of Hull (1988) - Theisis Title:
*Theoretical Predictions of Rime Ice Accretion and Snow Loading on Overhead Transmission Lines using Free Streamline Theory*

## Recent publications

I have authored and co-authored over 100 peer-reviewed publications, the majority of which are journal articles totalling 1,000+ pages. Recent publications (post millenium, at least) are as follows:

**2000**

- LARCOMBE, P.J. (2000) On Catalan Numbers and Expanding the Sine Function,
*Bulletin of the I.C.A*., 28, pp.39-47. - LARCOMBE, P.J. (2000) A Forgotten Convolution Type Identity of Catalan,
*Utilitas Mathematica*, 57, pp.65-72. - LARCOMBE, P.J. and FRENCH, D.R. (2000) On the `Other' Catalan Numbers: A Historical Formulation Re-Examined,
*Congressus Numerantium*, 143, pp.33-64.

**2001**

- LARCOMBE, P.J. and GESSEL, I.M. (2001) A Forgotten Convolution Type Identity of Catalan: Two Hypergeometric Proofs, Utilitas Mathematica, 59, pp.97-109.
- LARCOMBE, P.J. and FRENCH, D.R. (2001) On Expanding the Sine Function with Catalan Numbers: A Note on a Role for Hypergeometric Functions,
*Journal of Combinatorial Mathematics and Combinatorial Computing*, 37, pp.65-74. - LARCOMBE, P.J., FRENCH, D.R. and FENNESSEY, E.J. (2001) The Asymptotic Behaviour of the Catalan-Larcombe-French Sequence $\{ 1,8,80,896,10816, \ldots \}$,
*Utilitas Mathematica*, 60, pp.67-77. - LARCOMBE, P.J. and FRENCH, D.R. (2001) On the Integrality of the Catalan-Larcombe-French Sequence $\{ 1,8,80,896,10816, \ldots \}$,
*Congressus Numerantium*, 148, pp.65-91. - LARCOMBE, P.J. and WILSON, P.D.C. (2001) On the Generating Function of the Catalan Sequence: A Historical Perspective,
*Congressus Numerantium*, 149, pp.97-108.

**2002**

- LARCOMBE, P.J. (2002) On a New Formulation of Xinrong for the Embedding of Catalan Numbers in Series Forms of the Sine Function,
*Journal of Combinatorial Mathematics and Combinatorial Computing*, 42, pp.209-221. - LARCOMBE, P.J. and FRENCH, D.R. (2002) A New Proof of the Integral Form for the General Catalan Number Using a Trigonometric Identity of Bullard,
*Bulletin of the I.C.A*., 36, pp.37-45. - LARCOMBE, P.J., FRENCH, D.R. and WOODHAM, C.A. (2002) A Note on the Asymptotic Behaviour of a Prime Factor Decomposition of the General Catalan-Larcombe-French Number,
*Congressus Numerantium*, 156, pp.17-25. - LARCOMBE, P.J., FRENCH, D.R. and FENNESSEY, E.J. (2002) The Fennessey-Larcombe-French Sequence $\{ 1,8,144,2432,40000, \cdots \}$: Formulation and Asymptotic Form,
*Congressus Numerantium*, 158, pp.179-190.

**2003**

- LARCOMBE, P.J. (2003) On Bullard's 'Delta' Parameter: Properties of a Special Case,
*Bulletin of the I.C.A*., 37, pp.19-28. - LARCOMBE, P.J. and FRENCH, D.R. (2003) The Catalan Number $k$-Fold Self-Convolution Identity: The Original Formulation,
*Journal of Combinatorial Mathematics and Combinatorial Computing*, 46, pp.191-204. - LARCOMBE, P.J., FENNESSEY, E.J., KOEPF, W.A. and FRENCH, D.R. (2003) On Gould's Identity No. 1.45,
*Utilitas Mathematica*, 64, pp.19-24. - LARCOMBE, P.J., FRENCH, D.R. and GESSEL, I.M. (2003) On the Identity of von Szily: Original Derivation and a New Proof,
*Utilitas Mathematica*, 64, pp.167-181. - LARCOMBE, P.J., FRENCH, D.R. and FENNESSEY, E.J. (2003) The Fennessey-Larcombe-French Sequence $\{ 1,8,144,2432,40000, \cdots \}$: A Recursive Formulation and Prime Factor Decomposition,
*Congressus Numerantium*, 160, pp.129-137. - JARVIS, A.F., LARCOMBE, P.J. and FRENCH, D.R. (2003) Applications of the A.G.M. of Gauss: Some New Properties of the Catalan-Larcombe-French Sequence,
*Congressus Numerantium*, 161, pp.151-162. - LARCOMBE, P.J., FENNESSEY, E.J., KOEPF, W.A. and FRENCH, D.R. (2003) The Catalan Numbers Re-Visit the World Series,
*Congressus Numerantium*, 165, pp.19-32.

**2004**

- LARCOMBE, P.J., RIESE, A. and ZIMMERMANN, B. (2004) Computer Proofs of Matrix Product Identities,
*Journal of Algebra and its Applications*, 3, pp.105-109. - LARCOMBE, P.J., FENNESSEY, E.J. and KOEPF, W.A. (2004) Integral Proofs of Two Alternating Sign Binomial Coefficient Identities,
*Utilitas Mathematica*, 66, pp.93-103. - LARCOMBE, P.J. and FRENCH, D.R. (2004) A New Generating Function for the Catalan-Larcombe-French Sequence: Proof of a Result by Jovovic,
*Congressus Numerantium*, 166, pp.161-172. - JARVIS, A.F., LARCOMBE, P.J. and FRENCH, D.R. (2004) Linear Recurrences Between Two Recent Integer Sequences,
*Congressus Numerantium*, 169, pp.79-99.

**2005**

- JARVIS, A.F., LARCOMBE, P.J. and FRENCH, D.R. (2005) Power Series Identities Generated by Two Recent Integer Sequences,
*Bulletin of the I.C.A*., 43, pp.85-95. - LARCOMBE, P.J., LARSEN, M.E. and FENNESSEY, E.J. (2005) On Two Classes of Identities Involving Harmonic Numbers,
*Utilitas Mathematica*, 67, pp.65-80. - LARCOMBE, P.J. (2005) On Some Catalan Identities of Shapiro,
*Journal of Combinatorial Mathematics and Combinatorial Computing*, 54, pp.165-174. - LARCOMBE, P.J. (2005) A New Asymptotic Relation Between Two Recent Integer Sequences,
*Congressus Numerantium*, 175, pp.111-116. - JARVIS, A.F., LARCOMBE, P.J. and FRENCH, D.R. (2005) On Small Prime Divisibility of the Catalan-Larcombe-French Sequence,
*Indian Journal of Mathematics*, 47, pp.159-181.

**2006**

- LARCOMBE, P.J. (2006) Proof of a Hypergeometric Identity,
*Journal of Combinatorial Mathematics and Combinatorial Computing*, 57, pp.65-73. - JARVIS, A.F., LARCOMBE, P.J. and FRENCH, D.R. (2006) A Short Proof of the 2-Adic Valuation of the Catalan-Larcombe-French Number,
*Indian Journal of Mathematics*, 48, pp.135-138. - LARCOMBE, P.J. (2006) On Certain Series Expansions of the Sine Function Containing Embedded Catalan Numbers: A Complete Analytic Formulation,
*Journal of Combinatorial Mathematics and Combinatorial Computing*, 59, pp.3-16. - LARCOMBE, P.J. (2006) Formal Proofs of the Limiting Behaviour of Two Finite Series Using Dominated Convergence,
*Congressus Numerantium*, 178, pp.125-146. - LARCOMBE, P.J. (2006) A Generating Function for the Catalan-Larcombe-French Sequence via the Binomial Transform,
*Congressus Numerantium*, 181, pp.49-63.

**2007**

- KIRSCHENHOFER, P. and LARCOMBE, P.J. (2007) On a Class of Recursive-Based Binomial Coefficient Identities Involving Harmonic Numbers,
*Utilitas Mathematica*, 73, pp.105-115. - LARSEN, M.E. and LARCOMBE, P.J. (2007) Some Binomial Coefficient Identities of Specific and General Type,
*Utilitas Mathematica*, 74, pp.33-53. - LARCOMBE, P.J. (2007) On the Summation of a New Class of Infinite Series,
*Journal of Combinatorial Mathematics and Combinatorial Computing*, 60, pp.127-137. - LARCOMBE, P.J. (2007) An Elegant Hypergeometric Identity,
*Congressus Numerantium*, 184, pp.185-192.

**2008 **

- CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2008) On Iterated Generating Functions for Arbitrary Finite Sequences,
*Utilitas Mathematica*, 76, pp.115-128. - CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2008) On Iterated Generating Functions for Integer Sequences, and Catalan Polynomials,
*Utilitas Mathematica*, 77, pp.3-33. - CLAPPERTON, J.A., LARCOMBE, P.J., FENNESSEY, E.J. and LEVRIE, P. (2008) A Class of Auto-Identities for Catalan Polynomials, and Pad'{e} Approximation,
*Congressus Numerantium*, 189, pp.77-95. - LARCOMBE, P.J. and LARSEN, M.E. (2008) Dixon's Terminating $_{3}F_{2}(1)$: Proof of the Symmetric Form,
*Congressus Numerantium*, 192, pp.33-37.

**2009 **

- LARCOMBE, P.J. and LARSEN, M.E. (2009) The Sum $16^{n} \sum_{k=0}^{2n} 4^{k} { \frac{1}{2} \choose k }$ ${ -\frac{1}{2} \choose k }{ -2k \choose 2n-k }$: A Proof of its Closed Form,
*Utilitas Mathematica*, 79, pp.3-7. - KOEPF, W.A. and LARCOMBE, P.J. (2009) The Sum $16^{n} \sum_{k=0}^{2n} 4^{k} { \frac{1}{2} \choose k }$ ${ -\frac{1}{2} \choose k }{ -2k \choose 2n-k }$: A Computer Assisted Proof of its Closed Form, and Some Generalised Results,
*Utilitas Mathematica*, 79, pp.9-15. - CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2009) Some New Identities for Catalan Polynomials,
*Utilitas Mathematica*, 80, pp.3-10. - GESSEL, I.M. and LARCOMBE, P.J. (2009) The Sum $16^{n} \sum_{k=0}^{2n} 4^{k} { \frac{1}{2} \choose k }$ ${ -\frac{1}{2} \choose k }{ -2k \choose 2n-k }$: A Third Proof of its Closed Form,
*Utilitas Mathematica*, 80, pp.59-63.

**2010**

- CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2010) New Theory and Results from an Algebraic Application of Householder Root Finding Schemes,
*Utilitas Mathematica*, 83, pp.3-36. - LARCOMBE, P.J. and FRENCH, D.R. (2010) A New Catalan Convolution Identity,
*Congressus Numerantium*, 203, pp.193-211.

**2011**

- CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2011) On Iterated Generating Functions: A Class of Impossible Sequences,
*Utilitas Mathematica*, 84, pp.3-18. - CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2011) Two New Identities for Polynomial Families,
*Bulletin of the I.C.A*., 62, pp.25-32.

**2012**

- CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2012) New Closed Forms for Householder Root Finding Functions and Associated Non-Linear Polynomial Identities,
*Utilitas Mathematica*, 87, pp.131-150. - CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2012) The Delannoy Numbers: Three New Non-Linear Identities,
*Bulletin of the I.C.A*., 64, pp.39-56. - LARCOMBE, P.J. and FENNESSEY, E.J. (2012) Applying Integer Programming to Enumerate Equilibrium States of a Multi-Link Inverted Pendulum: A Strange Binomial Coefficient Identity and its Proof,
*Bulletin of the I.C.A*., 64, pp.83-108. - PARIS, R.B. and LARCOMBE, P.J. (2012) On the Asymptotic Expansion of a Binomial Sum Involving Powers of the Summation Index,
*Journal of Classical Analysis*, 1, pp.113-123. - LARCOMBE, P.J. and FENNESSEY, E.J. (2012) Some Properties of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$,
*Congressus Numerantium*, 214, pp.49-64.

**2013**

- LARCOMBE, P.J., BAGDASAR, O.D. and FENNESSEY, E.J. (2013) Horadam Sequences: A Survey,
*Bulletin of the I.C.A.*, 67, pp.49-72. - LARCOMBE, P.J. and FENNESSEY, E.J. (2013) On Iterated Generating Functions: A New Class of Lacunary 0-1 Impossible Sequences,
*Bulletin of the I.C.A.*, 67, pp.111-118. - BAGDASAR, O.D. and LARCOMBE, P.J. (2013) On the Characterization of Periodic Complex Horadam Sequences,
*Fibonacci Quarterly*, 51, pp.28-37. - LARCOMBE, P.J. and BAGDASAR, O.D. (2013) On a Result of Bunder Involving Horadam Sequences: A Proof and Generalization,
*Fibonacci Quarterly*, 51, pp.174-176. - BAGDASAR, O.D., LARCOMBE, P.J. and ANJUM, A. (2013) Particular Orbits of Periodic Horadam Sequences,
*Octogon Mathematics Magazine*, 21, pp.87-98. - BAGDASAR, O.D. and LARCOMBE, P.J. (2013) On the Number of Complex Horadam Sequences with a Fixed Period,
*Fibonacci Quarterly*, 51, pp.339-347.

**2014**

- KIRSCHENHOFER, P., LARCOMBE, P.J. and FENNESSEY, E.J. (2014) The Asymptotic Form of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$: Two Proofs,
*Utilitas Mathematica*, 93, pp.3-23. - CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2014) Generalised Catalan Polynomials and Their Properties,
*Bulletin of the I.C.A*., 71, pp.21-35. - JARVIS, A.F., LARCOMBE, P.J. and FENNESSEY, E.J. (2014) Some Factorisation and Divisibility Properties of Catalan Polynomials,
*Bulletin of the I.C.A*., 71, pp.36-56. - LARCOMBE, P.J. and FENNESSEY, E.J. (2014) On Cyclicity and Density of Some Catalan Polynomial Sequences,
*Bulletin of the I.C.A.*, 71, pp.87-93. - LARCOMBE, P.J. (2014) Closed Form Evaluations of Some Series Involving Catalan Numbers,
*Bulletin of the I.C.A.*, 71, pp.117-119. - LARCOMBE, P.J. and FENNESSEY, E.J. (2014) A Non-Linear Identity for a Particular Class of Polynomial Families,
*Fibonacci Quarterly*, 52, pp.75-79. - LARCOMBE, P.J., BAGDASAR, O.D. and FENNESSEY, E.J. (2014) On a Result of Bunder Involving Horadam Sequences: A New Proof,
*Fibonacci Quarterly*, 52, pp.175-177. - BAGDASAR, O.D. and LARCOMBE, P.J. (2014) On the Characterization of Periodic Generalized Horadam Sequences,
*J. Difference Equations and Applications*, 20, pp.1069-1090. - LARCOMBE, P.J., O'NEILL, S.T. and FENNESSEY, E.J. (2014) On Certain Series Expansions of the Sine Function: Catalan Numbers and Convergence,
*Fibonacci Quarterly*, 52, pp.236-242. - LARCOMBE, P.J. and FENNESSEY, E.J. (2014) Conditions Governing Cross-Family Member Equality in a Particular Class of Polynomial Families,
*Fibonacci Quarterly*, 52, pp.349-356.

**2015**

- LARCOMBE, P.J. and FENNESSEY, E.J. (2015) On Horadam Sequence Periodicity: A New Approach,
*Bulletin of the I.C.A.*, 73, pp.98-120. - LARCOMBE, P.J. and FENNESSEY, E.J. (2015) On the Phenomenon of Masked Periodic Horadam Sequences,
*Utilitas Mathematica*, 96, pp.111-123. - LARCOMBE, P.J. and FENNESSEY, E.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*, 53, pp.175-179. - LARCOMBE, P.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*, 53, pp.253-260. - JOHNSON, A., HOLMES, P., CRASKE, L., TROVATI, M., BESSIS, N. and LARCOMBE, P.J. (2015) Computational Objectivity in Depression Assessment for Unstructured Large Datasets,
*Proceedings of 13th I.E.E.E. International Conference on Dependable, Autonomic and Secure Computing*, Liverpool, U.K., October 26th-28th, pp.2075-2079. - TROVATI, M., TROVATI, J., LARCOMBE, P.J. and LIU, L. (2015) A Semi-Automated Assessment of the Direction of Influence Relations from Semantic Networks: A Case Study in Maths Anxiety,
*Proceedings of 13th I.E.E.E. International Conference on Dependable, Autonomic and Secure Computing*, Liverpool, U.K., October 26th-28th, pp.2088-2091. - LARCOMBE, P.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*, 53, pp.360-364.

**2016 **

- BAGDASAR, O.D., LARCOMBE, P.J. and ANJUM, A. (2016) On the Structure of Periodic Complex Horadam Orbits,
*Carpathian J. Mathematics*, 32, pp.29-36. - LARCOMBE, P.J. and RABAGO, J.F.T. (2016) On the Jacobsthal, Horadam and Geometric Mean Sequences,
*Bulletin of the I.C.A.*, 76, pp.117-126. - LARCOMBE, P.J. (2016) A New Formulation of a Result by McLaughlin for an Arbitrary Dimension 2 Matrix Power,
*Bulletin of the I.C.A.*, 76, pp.45-53. - LARCOMBE, P.J. and FENNESSEY, E.J. (2016) A Polynomial Based Construction of Periodic Horadam Sequences,
*Utilitas Mathematica*, 99, pp.231-239. - LARCOMBE, P.J. (2016) A Short Monograph on Exposition and the Emotive Nature of Research and Publishing,
*Mathematics Today*, 52, pp.86-90. - LARCOMBE, P.J. and FENNESSEY, E.J. (2016) On a Scaled Balanced-Power Product Recurrence,
*Fibonacci Quarterly*, 54, 242-246. - LARCOMBE, P.J. (2016) On the Evaluation of Sums of Exponentiated Multiples of Generalized Catalan Number Linear Combinations Using a Hypergeometric Approach,
*Fibonacci Quarterly*, 54, 259-270. - LARCOMBE, P.J. and FENNESSEY, E.J. (2016) A Scaled Power Product Recurrence Examined Using Matrix Methods,
*Bulletin of the I.C.A*, 78, 41-51. - LARCOMBE, P.J.
****(2016) Alwyn Francis Horadam, 1923-2016: A Personal Tribute to the Man and His Sequence,*Bulletin of the I.C.A*., 78, 93-107.

** ****2017 **

- LARCOMBE, P.J. and FENNESSEY, E.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 I.C.A*., to appear. - LARCOMBE, P.J. (2017) Opinions, Opinions, Opinions, ...,
*Mathematics Today,*53, 28-30. - LARCOMBE, P.J. and FENNESSEY, E.J. (2017) A Closed Form Formulation for the General Term of a Scaled Triple Power Product Recurrence Sequence,
*Fibonacci Quarterly*, to appear*.* - LARCOMBE, P.J. (2017) Horadam Sequences: A Survey Update and Extension,
*Bulletin of the I.C.A*., to appear. - LARCOMBE, P.J. (2017) Mathematics as a Mirror of Painting,
*Mathematics Today*, to appear.

## Recent conferences

- 24th British Combinatorial Conference, Royal Holloway, University of London, London, U.K., June 30th - July 5th, 2013
- 13th I.E.E.E. International Conference on Dependable, Autonomic and Secure Computing, Liverpool, U.K., October 26th-28th, 2015
- 1st I.M.A. Conference on Theoretical and Computational Discrete Mathematics, Derby, U.K., March 22nd-23rd, 2016. [ This was Derby University's first ever mathematics conference which I organised and chaired ]

## Additional interests and activities

I have been told that my life consists of family, mathematics, Aston Villa F.C., swimming (the prioritisation of which vary, depending on circumstances), and little else - I dispute this, however (and in any case this seems quite enough to me if I'm honest) !!

The students call me Professor Villa. My kids call me Mad Dad. My wife calls me various things. I call myself busy.

## In the media

I have written, and had solicited, informal pieces for the *Times Higher Education *on such things as academic snobbery in relation to personal accents, and levels of integrity within academia. Examples can be found below:

*Times Higher Education*Blog (4th June, 2016) : https://www.timeshighereducation.com/blog/im-professor-working-class-accent-get-over-it [ Follow up piece in the*Times Higher Education*, No. 2,260 (23rd June, 2016), p.35 ]*Times Higher Education*, No. 2,261 (30th June, 2016), p.39 [ see also https://www.timeshighereducation.com/features/is-there-a-problem-with-academic-integrity ]*Times Higher Education*, No. 2,263 (14th July, 2016), p.45 [ see also https://www.timeshighereducation.com/books/times-higher-education-summer-reads-2016 ]