About
In 2018 I completed a five year period overseeing the Mathematics programmes that we offered at undergraduate level during that time (BSc (Hons) in Mathematics, Mathematics and Computer Science, and Mathematics with Education); I continue to support or 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. I have an Erdős Number 2.
As well as 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 the BSc (Hons) in Computer Science and Mathematics), and at Liverpool John Moores University (September 2017 to June 2019 for the BA (Hons) in Mathematics and Education Studies); more recent activities include acting as the External Moderator for a syllabus design refresh across the suite of degrees in Mathematics at Nottingham Trent University (May 2023).
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 through a variety of metrics). 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 if the episode is true - 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 suggested 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. He added, "Let me then state it as an axiom that in order to be a good teacher, not just a popular one, one must have such an unwavering dedication and such an unbreakable commitment to one’s subject that intellectual somnambolism becomes unthinkable. But it is equally axiomatic that to be a good teacher one must be capable of joy and satisfaction in passing on the torch even if other hands made the fire that lit it. Render unto Caesar the things which are Caesar’s." So, what happened to the genuine all rounders? Out of fashion now it seems.
In the 1960s Polish born American mathematician Salomon Bochner wrote a detailed and personal appraisal of mathematics, covering its uniqueness as a force of our intellectuality, the mystique of its creativity, its then widening importance and growing efficacy, and its spread. He offered some engaging thoughts on the parallel emergence of mathematics and myths, through the device of 'symbolism', in the philosophies of Plato and his student Aristotle, writing "The similarity between mathematics and myths is grounded in a certain similarity of articulation; mathematics and myths both speak in "symbols", recognizably so, and the fact that it is nearly impossible to say satisfactorily, in or out of mathematics, what a symbol is does not destroy the similarity itself. Symbols in myths are very different from symbols in mathematics, but they are symbols all the same. Symbols in myths, as in poetry, may be charged, intentionally or half-intentionally, with ambivalences, whereas in mathematics they must not so be; nevertheless, even in myths symbols contribute a clarity and incisiveness which is peculiar to them, wherever they occur. Above all, even in myths symbolization creates the presumption that the verities which are proclaimed are endowed with a validity which is universal and unchanging, even if in mathematics the claim is much more pronounced and paramount than in myths. The potency of our mathematics today derives from the fact that its symbolization is cognitively logical and, what is decisive, operationally active and fertile; myths, however, from our retrospect, were always backward-directed, and their symbolizations were always reminiscingly anthropological and operationally inert." I think this is all rather interesting.
There can be a real joy when working with a colleague or within a small team, as ideas can be bounced off each other and mutual encouragement generated – sometimes the sum is greater than the individual parts, as they say. Working in solitude is great, too, as Albert Einstein recognised: "Bear in mind that those who are finer and nobler are always alone - and necessarily so - and that because of this they can enjoy the purity of their own atmosphere." He spoke thus, as early as his school days (when 16 years old, to be precise), highlighting a certain independence available in the scientific profession that greatly attracted him even then. Mathematicians appreciate the opportunity to work in isolation, more than most - I know I do.
It would be quite wrong to suppose that mathematicians feel obliged to limit their ideas and formulations to those with applications in the real world. That said, while pure mathematics can be fulfilling in the patterns and connections it reveals between constructs, the broader discipline does seem to offer a supreme and unique plan for understanding and mastering nature, and for this reason it garners much support. Morris Kline - mathematical philosopher, professor for over twenty years at New York University, and protagonist in the American curriculum reform in mathematics education that occurred during the second half of the 20th century - summed it up beautifully thus: "Mathematics may be the queen of the sciences and therefore entitled to royal prerogatives, but the queen who loses touch with her subjects may lose support and even be deprived of her realm. Mathematicians may like to rise into the clouds of abstract thought, but they should, and indeed they must, return to earth for nourishing food or else die of mental starvation. They are on safer and saner ground when they stay close to nature. As Wordsworth put it, "Wisdom oft is nearer when we stoop than when we soar.""
John E. Littlewood - long time collaborator with the great G.H. Hardy - produced a light-hearted essay in 1967 (republished posthumously 20 years later) which is worth a read. Musing on the mathematical life, he wrote "There is much to be said for being a mathematician. To begin with, he has to be completely honest in his work, not from any superior morality, but because he simply cannot get away with a fake. It has been cruelly said of arts dons, especially in Oxford, that they believe there is a polemical answer to everything; nothing is really true, and in controversy the object is to prove your opponent a fool. We escape all this. Further, the arts man is always on duty as a great mind; if he drops a brick, as we say in England, it reverberates down the years. After an honest day's work a mathematician goes off duty." He makes some good points here methinks.
Simon P. Norton - who died in 2019 aged 66 - worked with the great John Conway (1937-2020) at Cambridge on finite groups and thus cemented his place in the mathematical hierarchy as world class, though it came with many personal idiosyncracies. On 25th July, 1969, the Daily Mail published a short article on Norton, then a 17 year old Etonian student who had performed remarkably well for the UK team at the 11th International Mathematical Olympiad held in Bucharest earlier that month (he won a Gold Medal, and had done so in the previous two annual contests). On his return, when asked in interview if he ever felt isolated from his fellow human beings, he replied "That depends on what you mean. If you're asking whether mathematicians are isolated, then I think the answer is yes. But if the question is "Do they mind being isolated?" then I think the answer is no. At least I don't." Most people would agree that although it can occasionally be a physical reality, being removed from people is largely a state of mind which is most often a necessary (over and above a sufficient) condition to be a successful mathematician.
Israeli-American mathematician Doron Zeilberger is a bit of an iconoclast at Rutgers University in New Jersey. That's OK though. Forthright, opinionated, provocative, humorous and entertaining - and a fine mathematician in the world of discrete mathematics and computing (his areas of expertise include combinatorics, special functions, number theory, and (most notably) the role of computer-generated and computer-assisted proofs which he has championed relentlessly after producing ground-breaking work with Herbert S. Wilf in the 1990s) - he offered this view of being a mathematician during an interview in 2015: "Professional mathematicians, and more generally, all faculty members in universities and colleges, are very lucky that they can make (at least a modest) living by doing what they love to do, namely research, and that they would do, in their spare time, even if they did not get paid. Most of us professors have to teach a course or two per semester, and some of us do not like it, so for those people, it is a drawback. But many of us, myself included, love to teach, and share our love of math with our students, so this is also a great satisfaction (provided that the teaching load is not too high)." I wish there were more people like him around, adding real colour and energy to academic life - open, outspoken and unabashed maverick characters who have actually made significant contributions to the field of mathematics in both research and teaching never fail to impress.
French mathematician André Weil, taking stock of the field in a pensive essay not long after the end of World War 2, wrote " . . . , if logic is the hygiene of the mathematician, it is not his source of food; the great problems furnish the daily bread on which he thrives." Us work-a-day mathematicians don’t focus our energies on the 'great problems', but without doubt we feel Weil’s sentiment from those that do occupy our time.
In a 'round-the-table' discussion (published in June 2016) at the University of Tokyo's Kavli Institute for the Physics and Mathematics of the Universe, Princeton University's Israeli-American theoretical physicist Nathan Seiberg spoke of research in his field: "The reason research is interesting is because we're surprised by the answers. If we could predict the answers, we would not be surprised by them. Almost by definition, we cannot predict the outcome. So we should not attempt to do that. Pursue what you're interested in, keep working hard, pay attention to what's going on around you and be flexible - these are the rules. Sometimes it works, sometimes it doesn't." Precisely.
The prolific expositor Gian-Carlo Rota once wrote ''What makes a mathematician creative? Rule one: don't ask him to be creative. There is nothing deadlier for a mathematician than to be placed in a beautiful office and instructed to lay golden eggs. Creativity is never directly sought after. It comes indirectly... Creativity is a bad word. Unfortunately, we must leave it in the books because people in power believe in it. It is a dangerous word.'' I think that's fair enough.
Sometimes, when I am engaged in a period of intense study for research, my wife reminds me that I am not looking after myself properly - sufficient sleep (elusive at the best of times) becomes problematic, and I relegate food as a natural and almost unconscious act. This, I gather, is not a rare occurrence among we mathematicians, and it is expressed beautifully in Arthur Conan Doyle's short stories compilation The Case-Book of Sherlock Holmes. Holmes, deep in 'The Adventure of the Mazarin Stone', has his mind sharpened and concentration focused by the prospect of imminent death. Suggesting some relaxation with Dr. Watson before an anticipated murder attempt, he requests his customary tobacco and alcohol for such moments, commenting that the combination has to take the place of food. When Watson queries this, he explains thus: "Because the faculties become refined when you starve them. Why, surely, as a doctor, my dear Watson, you must admit that what your digestion gains in the way of blood supply is so much lost to the brain. I am a brain, Watson. The rest of me is a mere appendix. Therefore, it is the brain I must consider." Slightly dramatic, but what he is saying has more than a ring of truth in it for some.
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 Recurrence Sequences
After gaining a BSc Special Honours degree (1st Class) in Mathematics (University of Hull, 1984) and staying on to complete a PhD 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, USA), 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 (the first lying buried in an obscure journal article authored by the mathematician Eugene C. Catalan in the late 19^{th} century, the formulation of which I carefully worked through), 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 intensively 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 celebrated Catalan sequence, named after the mathematician himself. 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 Catalan was marked by a Special Issue (Vol. 76, May 2014) of the Bulletin of the ICA 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 PhD 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 IGF - 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 IGF 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 IGF 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. Having received news of the passing of Alwyn Horadam in July 2016 - just three or four years after I had started to take an interest in his work - I wrote and published a piece to honour both my own personal contact with him (by letter) and his endowment to the community of mathematicians; the essay 'Alwyn Francis Horadam, 1923-2016: A Personal Tribute to the Man and His Sequence' was very much appreciated by his family.
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 and publish accordingly. The great American linear algebraist Gil Strang was most surprised when he saw it (offering a proof himself for an article I published in 2015), and now includes it in talks as part of his invited lecture circuit. Some private discussions with the 1998 Fields Medalist Timothy Gowers has revealed that this simple anti-diagonals ratio is but one instance (actually the simplest) of an infinite number of matrix power invariants whose rational expressions are each first order and form a so called Class 1 group; the existence of other (infinite) groups with greater Class Number has been a surprise. The inimitable mathematician Doron Zeilberger has produced (2022) a new proof of 2 x 2 matrix power anti-diagonals ratio invariance, and discovered suites of complex invariant expressions (using the algebraic package Maple) for terms in exponentiated 3 x 3 and 4 x 4 matrices; he has also given a bijective proof of the said result for an n-square tri-diagonal matrix. Stronger yet identities involving mixed matrix powers have been found (using powerful elements of graph theory combined with ideas drawn from algebraic topology) by Mukherjee and Mukherjee for square matrices of arbitrary dimension (2023).