I have recently completed a four-year period overseeing the three Mathematics programmes that we offer at undergraduate level (BSc (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, 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 IMA (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 to June 2016 for Mathematics BSc/BA Combined (Hons) programmes), at Edge Hill University (September 2017 to June 2019 for BSc (Hons) in Computer Science and Mathematics), and at Liverpool John Moores University (September 2017 to June 2019 for BA (Hons) in Mathematics and Education Studies).
More interesting things to contemplate (or smile at)
An Amusement: 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 tongue-in-cheek 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 non-zero, 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.
The former Archbishop of Westminster, Cormac Murphy-O’Connor, spoke before he died in early September 2017 of what he saw as the continuing trend of both modern day secularisation and the marginalisation of faith, reportedly saying ''Religious belief of any kind tends to be treated more as a private eccentricity than as the central and formative element of British society that it is.'' Though mathematics is at the cornerstone of a range of overt and latent aspects of our world - lying behind many technological systems and artefacts that we use daily - it could be argued that its professional disciples suffer the same sort of discrimination. Mathematical votaries - whose common overriding telos is the discovery of different personal truths, and whose faith in this wondrous and at times mysterious discipline is unshakeable - are no less devoted than the most dedicated adherents of religion, but are thought of in similar kinds of ways that cause us to be dismissed as largely irrelevant and strange social outliers; this is, to me, both disappointing and odd in equal measure.
An Amusement: 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 co-authored papers) has the same ring to it as 'Hardy and Littlewood' - but, apart from matching syllables, that’s where the comparison ends !
Churning out paper after paper is, though a great temptation, neither a sufficient nor necessary condition for gaining respect in one's field of expertise - a mass of lightweight offerings amounts to little, while history tells us that some of those whose names stand out in mathematical lineage have sometimes produced but a few deep and insightful articles of distinction. We academics would do well to concentrate on the quality of our research rather than its volume, remembering always the advice of the Chinese sage Confucius who remarked "One should not be concerned at lack of position, but should be concerned about what will fit him to occupy it. One should not be concerned at being unknown, but should seek to be worthy of being known."
Sir Arthur Conan Doyle's great private detective character, Sherlock Holmes, describes his ratiocinative approach to investigation as being underpinned by the belief that the process "starts upon the supposition that when you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth." I find this to be very much the case, too, in mathematical research.
There seems to be a curious mix of febrility and anxiety in academia today - everything 'matters' so much, to everyone. People don't seem to want to be challenged any more, an overemotional tertiary sector appears to have lost its sense of humour somewhat, and genuine freedoms within research are becoming increasingly diminished - more importantly, the pressure to publish regularly is now a weight bearing down on us all, we compare ourselves to others constantly, and we are made to worry about our 'visibility' and 'impact' (as defined subjectively by others). As a slight play on words from the 1st Duke of Wellington (i.e., his riposte to blackmailer J.J. Stockdale (a pornographer and scandal-monger), who threatened to publish anecdotes of Wellington offered by his mistress Harriette Wilson and wanted money to ensure their omission (she was a famed British Regency courtesan who had 'formal relationship arrangements' with other significant politicians)), perhaps we have already moved to a new mantra for the 21st century: "Publish or be damned." Just a thought.
James Joyce - the Irish novelist, short story writer, poet, and occasional playwright and journalist – made a decision not to offer opinion in print about World War I (regarding politics and government as areas for the kinds of specialisms he did not have), but one of his biographers wrote that "He may not have gone to the battlefront but he was in the trenches with himself every day, . . ." Being a mathematician can feel a bit like that, for we constantly wrestle with the problems on which we work, and very often with ourselves during the process.
Us mathematicians have to be very committed to our research, and quite often it's an all-consuming occupation - that's just the way things are. According to Greek biographer and essayist Plutarch, the eminent scholar Archimedes was apparently so engaged in a mathematical problem that he didn't notice Roman forces (under General Marcus Claudius Marcellus) had broken through the Syracuse defence to take the city - he was killed by a soldier (who had apparently been tasked to take him to meet the commander) when he refused to move until he'd finished working on it. Now that is dedication - misplaced in this instance, of course, but indisputably genuine !
Polish-American mathematician Mark Kac wrote the following on classic academic personality types: "Let me be frank. A benevolent Mr. Chips who does nothing but teach freshmen calculus and hold the hand of every homesick student is as inadequate a teacher as the impatient, brilliant young expert in [some subject or other] who looks upon his teaching of freshman calculus as an indignity and a bore." I've known both kinds of colleague during my career, each able to look good as far as management are concerned, and tick their respective boxes of survival (the latter without the brilliance, unsurprisingly). Kac suggests that one uses teaching as a facade to hide mediocrity and ignorance, and the other conceals capriciousness and social irresponsibility under the popular banner of research. What happened to the genuine all rounders? Out of fashion now it seems.
Teaching responsibilities
I have delivered and/or led many different modules over the years at Derby, and have latterly been involved with the following undergraduate modules:
Linear Algebra (Year 2)
Mathematical Methods (Year 2)
Mathematics Group Project (Year 2)
Non-Linear Systems Dynamics (Year 3 and Masters level)
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 other contemporary and historical matters pertaining to mathematics. 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 a few works covering:
The role of exposition in mathematics
The emotive nature of research and publishing
Issues surrounding authorship and the effects of publishing pressures on individuals and the wider institutional workplace
Publishing standards in specific and general terms
The current move towards a rather dystopian higher education culture
My published works also take in other features of mathematical endeavour and its psychology, such as:
Links between mathematics and painting
The role of mathematical aesthetic in research and teaching
The notion of mathematical genius
The unsung writing abilities of some mathematicians
Aspects of getting older as a mathematician
The influence of past champions of mathematics within and outwith our mathematical community
I am happy to discuss possible PhD 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 non-linear) 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 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 continue to be examined by Chinese mathematicians looking at their convexity/concavity and congruency 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 might be relevant to the theory of automata); theoretical conditions under which new terms are added to an I.G.F. are also of interest, and have been looked at. 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 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 pleased with this 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 number theory and algebra with algorithmic computation, creating an underpinning theoretical framework to the concept of Horadam cyclicity with some beautiful visual results produced. In other work, periodic sequence 'masking' has been discovered, and further confirmed by appealing to properties of the Horadam sequence (ordinary) generating function. Recently, non-linear recurrence identitity classes for terms of the sequence (and others) have been found using new methodologies that have, where permissible, lent themselves to analogous results for higher order recursion systems. A new formulation of the well known closed form for the general Horadam sequence term has also been given, for both degenerate and non-degenerate characteristic roots, using matrix theory.
As an aside, a little known observation in linear algebra - that of the invariance of the anti-diagonals ratio with the power of a 2 x 2 matrix - has been proven in a variety of ways (and the result extended to describe invariance of all of the anti-diagonals ratios within an arbitrary dimension tri-diagonal matrix); this curiosity surprises people whenever they see it, and I continue to collect any new proofs I encounter.
Membership of professional bodies
Peter J. Larcombe : BSc, PhD, FIMA, CMath, CSci, MIET, CEng, FTICA, FHEA.
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 SEFI (European Society for Engineering Education)
I am a Chartered Engineer, Chartered Mathematician, and Chartered Scientist.
Qualifications
Undergraduate qualifications
BSc 1st Class (Special Honours) Degree in Mathematics, University of Hull (1984) [Awarded the 1984 University of Hull Slater Prize in Applied Mathematics]
Research qualifications
PhD 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 conferences
24th British Combinatorial Conference, Royal Holloway, University of London, U.K., 30 June to 5 July, 2013.
13th IEEE International Conference on Dependable, Autonomic and Secure Computing, Liverpool, UK, 26-28 October, 2015.
1st IMA Conference on Theoretical and Computational Discrete Mathematics, University of Derby, UK, 22-23 March, 2016. [This was the University of Derby's first ever mathematics conference which I organised and chaired]
2nd IMA Conference on Theoretical and Computational Discrete Mathematics, University of Derby, UK, 14-15 September, 2018. [Organised and chaired by myself]
Additional interests and activities
I serve on the Governors Board of the newly opened Derby Cathedral School, and I am an active member of the University of Derby Faith in Science Group.
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 just 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
the artificial promotion of staff, institutions and students as all 'winners'
I have authored and co-authored over 100 peer-reviewed publications, the majority of which are journal articles totalling 1,000+ pages. Recent publications are as follows:
LARCOMBE, P.J. (2021) Has Publishing Become a Pernicious Pursuit? On Culture, Compliance, Collaboration, Collusion, Control and Consequences, Palestine Journal of Mathematics, 10, to appear
LARCOMBE, P.J. (2020) Is the Fictional Dystopia of George Orwell's Novel Nineteen Eighty-Four Finally Coming to Pass as a New Quasi-Reality for U.K. Academe?, Palestine Journal of Mathematics, 9, pp.1-11
LARCOMBE, P.J. and FENNESSEY, E.J. (2020) On Anti-Diagonals Ratio Invariance with Exponentiation of a 2 x 2 Matrix: Two New Proofs, Palestine Journal of Mathematics, 9, pp.12-14
LARCOMBE, P.J. and FENNESSEY, E.J. (2020) A Formalised Inductive Approach to Establish the Invariance of Anti-Diagonal Ratios with Exponentiation for a Tri-Diagonal Matrix of Fixed Dimension, Palestine Journal of Mathematics, 9, to appear
LARCOMBE, P.J. and FENNESSEY, E.J. (2020) Preservation Conditions for Infinite Integer Sequence Iterated Generating Function Schemes, Palestine Journal of Mathematics, 9, to appear
LARCOMBE, P.J. and FENNESSEY, E.J. (2020) On Some Aspects of Horadam Sequence Periodicity via Generating Functions, Palestine Journal of Mathematics, 9, to appear
LARCOMBE, P.J. (2019) On the Notion of Mathematical Genius: Rhetoric and Reality, Palestine Journal of Mathematics, 8, pp.121-126
LARCOMBE, P.J. and FENNESSEY, E.J. (2019) On Generalised Multi-Index Non-Linear Recursion Identities for Terms of the Horadam Sequence, Palestine Journal of Mathematics, 8, pp.127-131
LARCOMBE, P.J. and FENNESSEY, E.J. (2019) New Proofs of Linear Recurrence Identities for Terms of the Horadam Sequence, Palestine Journal of Mathematics, 8, pp.132-137
LARCOMBE, P.J. and FENNESSEY, E.J. (2019) A Non-Linear Recurrence Identity Class for Terms of a Generalized Linear Recurrence Sequence of Degree Three, Fibonacci Quarterly, 57, pp.10-13
LARCOMBE, P.J. and FENNESSEY, E.J. (2019) A Four-Parameter Non-Linear Recurrence Identity Class for Terms of a Quasi Fibonacci Sequence, Palestine Journal of Mathematics, 8, pp.177-181
LARCOMBE, P.J. (2019) A Few Comments on Academic Publishing Standards in Relation to the Horadam Sequence and its Variants, Palestine Journal of Mathematics, 8, pp.320-323
O’NEILL, S.T. and LARCOMBE, P.J. (2019) A Note on the Closed Forms for the Horadam Sequence General Term, Palestine Journal of Mathematics, 8, pp.324-327
LARCOMBE, P.J. (2018) A Few Thoughts on the Aesthetics of Mathematics in Research and Teaching, Palestine Journal of Mathematics, 7, pp.1-8
LARCOMBE, P.J. and FENNESSEY, E.J. (2018) A New Tri-Diagonal Matrix Invariance Property, Palestine Journal of Mathematics, 7, pp.9-13
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.397-405
LARCOMBE, P.J. and FENNESSEY, E.J. (2018) A New Non-Linear Recurrence Identity Class for Horadam Sequence Terms, Palestine Journal of Mathematics, 7, pp.406-409
LARCOMBE, P.J. and FENNESSEY, E.J. (2018) A Note on Two Rational Invariants for a Particular 2 x 2 Matrix, Palestine Journal of Mathematics, 7, pp.410-413
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, pp.121-125
LARCOMBE, P.J. (2018) Mathematicians Can Also Write, Right?, Mathematics Today, 54, pp.56-58
LARCOMBE, P.J. and FENNESSEY, E.J. (2017) On Sequence-Based Closed Form Entries for an Exponentiated General 2 x 2 Matrix: A Re-Formulation and an Application, Bulletin of the I.C.A., 79, pp.82-94
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.168-177
LARCOMBE, P.J. (2017) Horadam Sequences: A Survey Update and Extension, Bulletin of the I.C.A., 80, pp.99-118
BAGDASAR, O.D. and LARCOMBE, P.J. (2017) On the Masked Periodicity of Horadam Sequences: A Generator-Based Approach, Fibonacci Quarterly, 55, pp.332-339
LARCOMBE, P.J. (2017) Mathematics as a Mirror of Painting, Mathematics Today, 53, pp.283-285
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, pp.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, pp.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, pp.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, pp.93-107
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 x 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 x 2 Matrix Anti-Diagonals Ratio with Increasing Matrix Power: Four Proofs, Fibonacci Quarterly, 53, pp.360-364
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
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
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
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
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
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