Hello, and welcome to my mathematics homepage. I am currently (June 2006 to June 2007) an Honorary Research Fellow at LaTrobe University Mathematics Department in Melbourne, Australia. I was a Visitor to the School of Mathematical Sciences at Australian National University Canberra, Australia in the 1990s, having completed a PhD there in 1998. In mid 2003 I started my association with La Trobe Maths Department and I have been slowly writing research in Number Theoretical areas.

During 2001 I completed a Graduate Diploma (Internet & Web Computing) at RMIT University in Melbourne. In the years 2002 to 2006 I have worked 18 months as an Analyst/Researcher for the Federal Government in the Attorney-General's Department, then three years with the Salvation Army Employment Plus as their Business Analyst at the National level reporting to the CEO.

As a mathematician, I have written 27 research papers, of which most are Number Theory, but with connections to Statistical Mechanics, Optics and Fractals or Self-similarity. Recently there has been a connection between my work and work on quasicrystals.

Also, during my researches, I discovered Dirichlet series analogues for q-series which enable me to write down new Dirichlet series summation formulas which are demonstrably analogues for known basic hypergeometric series summations. In other words, I found results which are similar in appearance to the well-known q-series identities such as the q-binomial theorem, the Heine q-series analogue of the Gauss hypergeometric sum, and analogues to Dixon's theorem. Also, in many instances, the well-known tranformations of q-series can become transformations of the new Dirichlet series. Curiously, I had the idea for these analogues in the early 1990s but it has taken until 2006 to get the first paper into print, for that see

CAMPBELL, G. B. An Euler Product transform applied to q series, Ramanujan J (2006) 12:267-293.

In recent years I also discovered a new class of infinite products which I knew had relevance to Number Theory. Then as the work developed I found that there were many areas of mathematics which connected.

A rather nice example of the new class of identities identities I found is

We use the notation (a, b, c, d, e) to mean the greatest common divisor of the integers a, b, c, d and e.

The introductory chapter to my thesis brings together the two main areas of my research in the context of known work. Right now I would like an interesting job either as an academic researcher or in the public sector as an executive administrator of some kind. I attach my Curriculum Vitae here accordingly.

My research interests are based on:-

• Number Theory
• Combinatorial Identities.
• Basic Hypergeometric Series, known also as q-series.
• Riemann's Zeta Function and its variants.
• Arithmetical Functions such as Divisor Functions.
• Gap power series with self-similar oscillations. These oscillations are related to chaos theory & fractals generated by simple recursions. Examples of fractals are:-
  
  These applets were obtained from the Yukie free applet page.

My favourite mathematics was or is being done by:-

• George E Andrews. Professor Andrews, of Pennsylvania State University, is a leading figure among Number Theorists and has a wealth of knowledge of q-series and their applications. He has written many wonderful papers and books.
• Richard Askey. Professor Askey is an expert on Hypergeometric Series as they apply to the theory of Special Functions. In particular, he has found q-integrals as for analogues for the Beta Function and also for the more general Selberg Integral.
• Rodney J Baxter. Professor Baxter is the Chair of Theoretical Physics at Australian National University. He is an expert in Statistical Mechanics, and has written a book on this subject. During the 1980s he expounded proofs of the Rogers-Ramanujan identities and similar results previously only obtained by Number Theorists. What is so special about this is that he did it entirely from a Statistical Mechanics perspective and completely independent of the known theories.
• G H Hardy (1877-1947). Godfrey Harold Hardy was a foremost figure in Pure Mathematics during the first half of the 20th century. He wrote many papers and books which are of lasting significance to Number Theorists and was responsible for the "discovery" of the Indian mathematician Srinivasa Ramanujan.
• Alf van der Poorten. (Macquarie University Sydney) Professor van der Poorten wrote a fascinating book about Fermat's Last Theorem; the first such book embracing the long history of the subject since the theorem was proved by Andrew Wiles in recent years.

I share with you a treasured photograph of myself and Professor Baxter on my graduation day, April 17, 1998.