Professor Peter J Larcombe
Position: Professor of Discrete and Applied Mathematics
College: College of Engineering and Technology
Department: Electronics, 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 (visiting schools/colleges to talk about studying mathematics and offering 'taster' sessions).
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/discoveringbeautymathsgainingkeyskills/story19659407detail/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), of which I have been a longstanding member.
I acted as External Examiner at Liverpool Hope University (September 2012  June 2016; Mathematics B.Sc./B.A. Combined (Hons.) Programmes), and am currently External Examiner at Edge Hill University (September 2017 onwards; B.Sc. (Hons.) in Computer Science and Mathematics), and at Liverpool John Moores University (September 2017 onwards; B.A. (Hons.) in Mathematics and Education Studies).
Alfréd Rényi, a colleague and collaborator of the great Hungarian mathematician Paul Erdős, is reputed to have said something like "A mathematician is a machine for turning coffee into theorems" (it was originally intended as a tongueincheek remark to explain the serious coffee meeting culture that flourished among Hungarian mathematicians at the time and produced some major advances in the field; to mathematicians, a coffee meeting is a great chance to do some mathematics !). I drink stacks of coffee but, though nonzero, my ratio of theorems to cups of coffee remains disappointingly small . . . .
Russian author Lev Nikolayevich Tolstoy was reported as saying something along the lines of the following: "A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator, the smaller the fraction." How many people who fractionally rank as < 1 do you know ? I know plenty unfortunately. I like to think my ranking never falls below unity.
A Muse: I and my longstanding research partner and friend, Eric J. Fennessey, sometimes joke between us that 'Larcombe and Fennessey' (as we always use on our coauthored papers) has the same ring to it as 'Hardy and Littlewood' – but, apart from matching syllables, that’s where the comparison ends !
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)
 NonLinear 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 works covering the role of exposition in mathematics, the emotive nature of research and publishing, issues surrounding authorship, and other features of mathematical endeavour and psychology  such as links between mathematics and painting, the role of mathematical aesthetic in research and teaching, the notion of mathematical genius, and the unsung writing abilities of some mathematicians.
I am happy to discuss possible Ph.D. supervisions with potential students interested in any of the following broad topics:
 The issue of mathematical anxiety in students and teachers
 Theoretical and applicative aspects of (linear and nonlinear) recurrence sequences
 The theory of iterated generating functions
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 NonLinear 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 CatalanLarcombeFrench and FennesseyLarcombeFrench 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 nonlinear 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 OnLine 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 CLF and FLF 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 NewtonRaphson and Halley rootfinding versions prominent in numerical analysis), then its algebraic execution by computer generates a pair of nonlinear 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 spinoff 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 reexamined and new results found. Horadam sequences are secondorder 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, PellLucas, Jacobsthal, JacobsthalLucas, Tagiuri, Fermat, FermatLucas  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, selfrepeating) sequences and identified necessary and sufficient conditions governing periodicity of complex Horadam sequences under general initial values, offering a full characterisation (of degenerate and nondegenerate 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
Peter J. Larcombe : B.Sc., Ph.D., F.I.M.A., C.Math., C.Sci., M.I.E.T., C.Eng., F.T.I.C.A., F.H.E.A.
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
 Fellow of the Higher Education Academy
 Member of the Institute 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)  Thesis Title: Theoretical Predictions of Rime Ice Accretion and Snow Loading on Overhead Transmission Lines using Free Streamline Theory
Recent publications
I have authored and coauthored over 100 peerreviewed 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.3947.
 LARCOMBE, P.J. (2000) A Forgotten Convolution Type Identity of Catalan, Utilitas Mathematica, 57, pp.6572.
 LARCOMBE, P.J. and FRENCH, D.R. (2000) On the `Other' Catalan Numbers: A Historical Formulation ReExamined, Congressus Numerantium, 143, pp.3364.
2001
 LARCOMBE, P.J. and GESSEL, I.M. (2001) A Forgotten Convolution Type Identity of Catalan: Two Hypergeometric Proofs, Utilitas Mathematica, 59, pp.97109.
 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.6574.
 LARCOMBE, P.J., FRENCH, D.R. and FENNESSEY, E.J. (2001) The Asymptotic Behaviour of the CatalanLarcombeFrench Sequence $\{ 1,8,80,896,10816, \ldots \}$, Utilitas Mathematica, 60, pp.6777.
 LARCOMBE, P.J. and FRENCH, D.R. (2001) On the Integrality of the CatalanLarcombeFrench Sequence $\{ 1,8,80,896,10816, \ldots \}$, Congressus Numerantium, 148, pp.6591.
 LARCOMBE, P.J. and WILSON, P.D.C. (2001) On the Generating Function of the Catalan Sequence: A Historical Perspective, Congressus Numerantium, 149, pp.97108.
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.209221.
 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.3745.
 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 CatalanLarcombeFrench Number, Congressus Numerantium, 156, pp.1725.
 LARCOMBE, P.J., FRENCH, D.R. and FENNESSEY, E.J. (2002) The FennesseyLarcombeFrench Sequence $\{ 1,8,144,2432,40000, \cdots \}$: Formulation and Asymptotic Form, Congressus Numerantium, 158, pp.179190.
2003
 LARCOMBE, P.J. (2003) On Bullard's 'Delta' Parameter: Properties of a Special Case, Bulletin of the I.C.A., 37, pp.1928.
 LARCOMBE, P.J. and FRENCH, D.R. (2003) The Catalan Number $k$Fold SelfConvolution Identity: The Original Formulation, Journal of Combinatorial Mathematics and Combinatorial Computing, 46, pp.191204.
 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.1924.
 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.167181.
 LARCOMBE, P.J., FRENCH, D.R. and FENNESSEY, E.J. (2003) The FennesseyLarcombeFrench Sequence $\{ 1,8,144,2432,40000, \cdots \}$: A Recursive Formulation and Prime Factor Decomposition, Congressus Numerantium, 160, pp.129137.
 JARVIS, A.F., LARCOMBE, P.J. and FRENCH, D.R. (2003) Applications of the A.G.M. of Gauss: Some New Properties of the CatalanLarcombeFrench Sequence, Congressus Numerantium, 161, pp.151162.
 LARCOMBE, P.J., FENNESSEY, E.J., KOEPF, W.A. and FRENCH, D.R. (2003) The Catalan Numbers ReVisit the World Series, Congressus Numerantium, 165, pp.1932.
2004
 LARCOMBE, P.J., RIESE, A. and ZIMMERMANN, B. (2004) Computer Proofs of Matrix Product Identities, Journal of Algebra and its Applications, 3, pp.105109.
 LARCOMBE, P.J., FENNESSEY, E.J. and KOEPF, W.A. (2004) Integral Proofs of Two Alternating Sign Binomial Coefficient Identities, Utilitas Mathematica, 66, pp.93103.
 LARCOMBE, P.J. and FRENCH, D.R. (2004) A New Generating Function for the CatalanLarcombeFrench Sequence: Proof of a Result by Jovovic, Congressus Numerantium, 166, pp.161172.
 JARVIS, A.F., LARCOMBE, P.J. and FRENCH, D.R. (2004) Linear Recurrences Between Two Recent Integer Sequences, Congressus Numerantium, 169, pp.7999.
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.8595.
 LARCOMBE, P.J., LARSEN, M.E. and FENNESSEY, E.J. (2005) On Two Classes of Identities Involving Harmonic Numbers, Utilitas Mathematica, 67, pp.6580.
 LARCOMBE, P.J. (2005) On Some Catalan Identities of Shapiro, Journal of Combinatorial Mathematics and Combinatorial Computing, 54, pp.165174.
 LARCOMBE, P.J. (2005) A New Asymptotic Relation Between Two Recent Integer Sequences, Congressus Numerantium, 175, pp.111116.
 JARVIS, A.F., LARCOMBE, P.J. and FRENCH, D.R. (2005) On Small Prime Divisibility of the CatalanLarcombeFrench Sequence, Indian Journal of Mathematics, 47, pp.159181.
2006
 LARCOMBE, P.J. (2006) Proof of a Hypergeometric Identity, Journal of Combinatorial Mathematics and Combinatorial Computing, 57, pp.6573.
 JARVIS, A.F., LARCOMBE, P.J. and FRENCH, D.R. (2006) A Short Proof of the 2Adic Valuation of the CatalanLarcombeFrench Number, Indian Journal of Mathematics, 48, pp.135138.
 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.316.
 LARCOMBE, P.J. (2006) Formal Proofs of the Limiting Behaviour of Two Finite Series Using Dominated Convergence, Congressus Numerantium, 178, pp.125146.
 LARCOMBE, P.J. (2006) A Generating Function for the CatalanLarcombeFrench Sequence via the Binomial Transform, Congressus Numerantium, 181, pp.4963.
2007
 KIRSCHENHOFER, P. and LARCOMBE, P.J. (2007) On a Class of RecursiveBased Binomial Coefficient Identities Involving Harmonic Numbers, Utilitas Mathematica, 73, pp.105115.
 LARSEN, M.E. and LARCOMBE, P.J. (2007) Some Binomial Coefficient Identities of Specific and General Type, Utilitas Mathematica, 74, pp.3353.
 LARCOMBE, P.J. (2007) On the Summation of a New Class of Infinite Series, Journal of Combinatorial Mathematics and Combinatorial Computing, 60, pp.127137.
 LARCOMBE, P.J. (2007) An Elegant Hypergeometric Identity, Congressus Numerantium, 184, pp.185192.
2008
 CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2008) On Iterated Generating Functions for Arbitrary Finite Sequences, Utilitas Mathematica, 76, pp.115128.
 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.333.
 CLAPPERTON, J.A., LARCOMBE, P.J., FENNESSEY, E.J. and LEVRIE, P. (2008) A Class of AutoIdentities for Catalan Polynomials, and Pad'{e} Approximation, Congressus Numerantium, 189, pp.7795.
 LARCOMBE, P.J. and LARSEN, M.E. (2008) Dixon's Terminating $_{3}F_{2}(1)$: Proof of the Symmetric Form, Congressus Numerantium, 192, pp.3337.
2009
 LARCOMBE, P.J. and LARSEN, M.E. (2009) The Sum $16^{n} \sum_{k=0}^{2n} 4^{k} { \frac{1}{2} \choose k }$ ${ \frac{1}{2} \choose k }{ 2k \choose 2nk }$: A Proof of its Closed Form, Utilitas Mathematica, 79, pp.37.
 KOEPF, W.A. and LARCOMBE, P.J. (2009) The Sum $16^{n} \sum_{k=0}^{2n} 4^{k} { \frac{1}{2} \choose k }$ ${ \frac{1}{2} \choose k }{ 2k \choose 2nk }$: A Computer Assisted Proof of its Closed Form, and Some Generalised Results, Utilitas Mathematica, 79, pp.915.
 CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2009) Some New Identities for Catalan Polynomials, Utilitas Mathematica, 80, pp.310.
 GESSEL, I.M. and LARCOMBE, P.J. (2009) The Sum $16^{n} \sum_{k=0}^{2n} 4^{k} { \frac{1}{2} \choose k }$ ${ \frac{1}{2} \choose k }{ 2k \choose 2nk }$: A Third Proof of its Closed Form, Utilitas Mathematica, 80, pp.5963.
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.336.
 LARCOMBE, P.J. and FRENCH, D.R. (2010) A New Catalan Convolution Identity, Congressus Numerantium, 203, pp.193211.
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.318.
 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.2532.
2012
 CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2012) New Closed Forms for Householder Root Finding Functions and Associated NonLinear Polynomial Identities, Utilitas Mathematica, 87, pp.131150.
 CLAPPERTON, J.A., LARCOMBE, P.J. and FENNESSEY, E.J. (2012) The Delannoy Numbers: Three New NonLinear Identities, Bulletin of the I.C.A., 64, pp.3956.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2012) Applying Integer Programming to Enumerate Equilibrium States of a MultiLink Inverted Pendulum: A Strange Binomial Coefficient Identity and its Proof, Bulletin of the I.C.A., 64, pp.83108.
 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.113123.
 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.4964.
2013
 LARCOMBE, P.J., BAGDASAR, O.D. and FENNESSEY, E.J. (2013) Horadam Sequences: A Survey, Bulletin of the I.C.A., 67, pp.4972.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2013) On Iterated Generating Functions: A New Class of Lacunary 01 Impossible Sequences, Bulletin of the I.C.A., 67, pp.111118.
 BAGDASAR, O.D. and LARCOMBE, P.J. (2013) On the Characterization of Periodic Complex Horadam Sequences, Fibonacci Quarterly, 51, pp.2837.
 LARCOMBE, P.J. and BAGDASAR, O.D. (2013) On a Result of Bunder Involving Horadam Sequences: A Proof and Generalization, Fibonacci Quarterly, 51, pp.174176.
 BAGDASAR, O.D., LARCOMBE, P.J. and ANJUM, A. (2013) Particular Orbits of Periodic Horadam Sequences, Octogon Mathematics Magazine, 21, pp.8798.
 BAGDASAR, O.D. and LARCOMBE, P.J. (2013) On the Number of Complex Horadam Sequences with a Fixed Period, Fibonacci Quarterly, 51, pp.339347.
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.323.
 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.2135.
 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.3656.
 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.8793.
 LARCOMBE, P.J. (2014) Closed Form Evaluations of Some Series Involving Catalan Numbers, Bulletin of the I.C.A., 71, pp.117119.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2014) A NonLinear Identity for a Particular Class of Polynomial Families, Fibonacci Quarterly, 52, pp.7579.
 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.175177.
 BAGDASAR, O.D. and LARCOMBE, P.J. (2014) On the Characterization of Periodic Generalized Horadam Sequences, J. Difference Equations and Applications, 20, pp.10691090.
 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.236242.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2014) Conditions Governing CrossFamily Member Equality in a Particular Class of Polynomial Families, Fibonacci Quarterly, 52, pp.349356.
2015
 LARCOMBE, P.J. and FENNESSEY, E.J. (2015) On Horadam Sequence Periodicity: A New Approach, Bulletin of the I.C.A., 73, pp.98120.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2015) On the Phenomenon of Masked Periodic Horadam Sequences, Utilitas Mathematica, 96, pp.111123.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2015) A Condition for AntiDiagonals Product Invariance Across Powers of $2 \times 2$ Matrix Sets Characterizing a Particular Class of Polynomial Families, Fibonacci Quarterly, 53, pp.175179.
 LARCOMBE, P.J. (2015) Closed Form Evaluations of Some Series Comprising Sums of Exponentiated Multiples of TwoTerm and ThreeTerm Catalan Number Linear Combinations, Fibonacci Quarterly, 53, pp.253260.
 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 26th28th, pp.20752079.
 TROVATI, M., TROVATI, J., LARCOMBE, P.J. and LIU, L. (2015) A SemiAutomated 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 26th28th, pp.20882091.
 LARCOMBE, P.J. (2015) A Note on the Invariance of the General $2 \times 2$ Matrix AntiDiagonals Ratio with Increasing Matrix Power: Four Proofs, Fibonacci Quarterly, 53, pp.360364.
2016
 BAGDASAR, O.D., LARCOMBE, P.J. and ANJUM, A. (2016) On the Structure of Periodic Complex Horadam Orbits, Carpathian J. Mathematics, 32, pp.2936.
 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.117126.
 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.4553.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2016) A Polynomial Based Construction of Periodic Horadam Sequences, Utilitas Mathematica, 99, pp.231239.
 LARCOMBE, P.J. (2016) A Short Monograph on Exposition and the Emotive Nature of Research and Publishing, Mathematics Today, 52, pp.8690.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2016) On a Scaled BalancedPower Product Recurrence, Fibonacci Quarterly, 54, pp.242246.
 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, pp.259270.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2016) A Scaled Power Product Recurrence Examined Using Matrix Methods, Bulletin of the I.C.A, 78, pp.4151.
 LARCOMBE, P.J. (2016) Alwyn Francis Horadam, 19232016: A Personal Tribute to the Man and His Sequence, Bulletin of the I.C.A., 78, pp.93107.
2017
 LARCOMBE, P.J. and FENNESSEY, E.J. (2017) On SequenceBased Closed Form Entries for an Exponentiated General $2 \times 2$ Matrix: A ReFormulation and an Application, Bulletin of the I.C.A., 79, pp.8294.
 LARCOMBE, P.J. (2017) Opinions, Opinions, Opinions ..., Mathematics Today, 53, pp.2830.
 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, 55, pp.168177.
 LARCOMBE, P.J. (2017) Horadam Sequences: A Survey Update and Extension, Bulletin of the I.C.A., 80, pp.99118.
 BAGDASAR, O.D. and LARCOMBE, P.J. (2017) On the Masked Periodicity of Horadam Sequences: A GeneratorBased Approach, Fibonacci Quarterly, 55, pp.332339.
 LARCOMBE, P.J. (2017) Mathematics as a Mirror of Painting, Mathematics Today, 53, pp.283285.
2018
 LARCOMBE, P.J. (2018) A Few Thoughts on the Aesthetics of Mathematics in Research and Teaching, Palestine Journal of Mathematics, 7, pp.18.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2018) A New TriDiagonal Matrix Invariance Property, Palestine Journal of Mathematics, 7, pp.913.
 LARCOMBE, P.J., RABAGO, J.F.T. and FENNESSEY, E.J. (2018) On Two Derivative Sequences from Scaled Geometric Mean Sequence Terms, Palestine Journal of Mathematics, 7, pp.397405.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2018) A New NonLinear Recurrence Identity Class for Horadam Sequence Terms, Palestine Journal of Mathematics, 7, pp.406409.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2018) A Note on Two Rational Invariants for a Particular $2 \times 2$ Matrix, Palestine Journal of Mathematics, 7, pp.410413.
 LARCOMBE, P.J. and O'NEILL, S.T. (2018) A Generating Function Approach to the Automated Evaluation of Sums of Exponentiated Multiples of Generalized Catalan Number Linear Combinations, Fibonacci Quarterly, 56, to appear.
 LARCOMBE, P.J. (2018) Mathematicians Can Also Write, Right?, Mathematics Today, 54, pp.5658.
 LARCOMBE, P.J. (2018) On the Notion of Mathematical Genius: Rhetoric and Reality, Palestine Journal of Mathematics, to appear.
2019
 LARCOMBE, P.J. (2019) When 'Smoke and Mirrors' Permeate Publishing, We Have a Problem, Palestine Journal of Mathematics, to appear.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2019) On Generalised MultiIndex NonLinear Recursion Identities for Terms of the Horadam Sequence, Palestine Journal of Mathematics, in preparation.
 LARCOMBE, P.J. and FENNESSEY, E.J. (2019) A NonLinear Recurrence Identity Class for Terms of a Generalized Linear Recurrence Sequence of Degree Three, Fibonacci Quarterly, in preparation.
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 26th28th, 2015
 1st I.M.A. Conference on Theoretical and Computational Discrete Mathematics, Derby, U.K., March 22nd23rd, 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, time, mood, and so on), and little else  I kind of dispute this, but in any case this seems quite enough to me if I'm honest.
Some of 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
 levels of integrity within academia
 the pervasive influence of overpraise and hype in schools and academia
Links and information are listed below:
 Times Higher Education Blog (4th June, 2016)  https://www.timeshighereducation.com/blog/improfessorworkingclassaccentgetoverit [ 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/isthereaproblemwithacademicintegrity ]
 Times Higher Education, No. 2,263 (14th July, 2016), p.45 [ see also https://www.timeshighereducation.com/books/timeshighereducationsummerreads2016 ]
 Times Higher Education Blog (22nd February, 2017)  https://www.timeshighereducation.com/blog/overpraiseeverywhereanduniversitiesarenotimmune