# Geoffrey Campbell

## Contacts

### Research interests

In 2022 I have written four "Fun with Numbers" brief articles for the Gazette of the Australian Mathematical Society.

My main mathematical preoccupation is with the below book project. My 2019 paper's approach to vector partition theory (See https://arxiv.org/abs/1906.07526) is the starting place for the project here; a monograph now entitled ** Partitions, Visible Point Vectors and Ramanujan Functions**. My motivation was to cover classical Integer Partitions, Rogers-Ramanujan Partitions in Statistical Mechanics Solved Models, and include Vector Partitions ideas and the

*Visible Point Vector identities*I found some years ago. However, it now has chapters on Plane Partitions, Asymptotic Partition Formulas, Partition Congruences, Ramanujan Continued Fractions, Polylogarithms, Parametric Euler sum identities, Higher Dimensional Weighted Partition Identities. I expect the final draft will be done by March 2023.

My high level table of contents, as at November 2022, is:

**Partitions, Visible Point Vectors and Ramanujan Functions**

Preface

1. About this book and its academic context

2. The author’s vision and motivations

Chapter 1. Historical background - The range of literature

1. Partitions as grown from the Ramanujan works

2. Andrews’ ”The Theory of Partitions” and ”Integer Partitions” with Eriksson

3. The 2021 book ”G. E. Andrews 80 Years of Combinatory Analysis”

4. Basic Hypergeometric Series

5. Lattice Sums in Chemistry molecular structures

6. Polylogarithms and computational research related results

7. Partition theory in Statistical Mechanics and Theoretical Physics

Chapter 2. A brief history and timeline for partitions

1. Leonhard Euler - 18th century

2. Gauss, Cauchy and Heine - 19th century

3. Rogers and Ramanujan - 1890s to 1920s

4. Major Percy MacMahon - 1896 to circa 1920

5. Hardy and Ramanujan - circa 1918 to the 1940s

6. Andrews - early 1960s to circa 2020

Chapter 3. Integer partitions and their generating functions

1. Preview

2. Euler’s approach to partitions

3. Euler’s partition identity

4. Euler’s pentagonal number theorem

5. The q-binomial Theorem

6. The Jacobi Triple Product

7. Two Identities from Gauss

8. The Watson Quintuple Product

9. The Heine fundamental transformation

10. The q-analogues of Gauss’s Theorem and Kummer’s Theorem

11. Euler-Pair theorem

12. Rogers-Ramanujan identities

Chapter 4. Continued Fractions for partitions generating functions

1. Euler’s Continued Fraction

2. Euler’s continued fraction applied to partitions.

3. Rogers-Ramanujan Continued Fractions for partition functions

4. Ramanujan’s three parameter continued fraction

Chapter 5. Congruence properties of partitions

1. Ramanujan’s partition congruence insights

2. Rodseth–Gupta theorem on binary partitions

3. The 2D binary partitions congruence theory

Chapter 6. Ferrers graphs and Ferrers Boards

1. Introduction

2. Conjugate Partitions

3. Ferrers Graphs and Ferrers Boards

4. Bressoud’s Bijection proof of Rogers-Ramanujan Identities

5. Franklin’s near-Bijection proof of Euler’s Pentagonal Number Theorem

Chapter 7. Durfee Squares

1. Introduction

2. Durfee Squares and Generating Functions

3. Frobenius Symbols

4. Jacobi’s Triple Product identity

5. The Rogers Ramanujan Identities

6. Successive Durfee Squares

Chapter 8. Gaussian polynomials

1. Definition and particular cases

2. Combinatorial interpretations

2.1. Balls into bins

2.2. Reflection

3. Analogs of Pascal’s identity

4. q-binomial theorem analogue

5. Gaussian polynomials in the Theory of Partitions

Chapter 9. Plane Partitions from MacMahon to Andrews

1. Definitions of Plane Partitions

2. Lozenge tilings of a hexagon.

3. Generating functions for Plane Partitions

4. The Ten Symmetry Classes of Plane Partitions

5. Class 1: Unrestricted Plane Partitions.

6. Class 2: Symmetric Plane Partitions.

7. Class 3: Cyclically Symmetric Plane Partitions.

8. Class 4: Totally Symmetric Plane Partitions.

9. Class 5: Self-Complementary Plane Partitions.

10. Class 6: Transpose-Complementary Plane Partitions.

11. Class 7: Symmetric Self-Complementary Plane Partitions.

12. Class 8: Cyclically Symmetric Transpose-Complementary Plane Partitions.

13. Class 9: Cyclically Symmetric Self-Complementary Plane Partitions.

14. Class 10: Totally Symmetric Self-Complementary Plane Partitions.

15. The State of Play circa 2022 and Open Questions

16. The Gog and Magog Trapezoids and Alternating Sign Matrices

17. Inferences from Plane Partitions that may apply to Vector Partitions

Chapter 10. Asymptotics for partition functions

1. The Circle Method of Hardy and Ramanujan

2. The Rademacher exact formula

3. The Theorem of Meinardus

4. A polynomial analogue of Meinardus’ Theorem

5. The Convolution Method for products of two series

6. Bruinier and Ono’s exact formula for p(n)

Chapter 11. The partition function in Statistical Mechanics

1. Baxter’s Hard Hexagon Model solved exactly by Rogers-Ramanujan identities

2. Hard Hexagon Model Regime II identities from Baxter

3. Outline of proofs of Baxter’s Regime II Conjectures

4. Transforms between Baxter’s Regime II and Regime III.

5. The partition function of the Hard Hexagon Model

5.1. Solution

6. Rogers-Ramanujan transition from Mathematics to Physics

Chapter 12. Vector partitions and their generating function identities

1. Defining Vector Grids and resulting Partition Grids

2. A theorem for 2D partitions into exactly two parts.

3. Visible Point Vector partitions and their generating functions

Chapter 13. Integer Partitions to Vector Partitions

1. Introduction

2. Setting up a higher dimensional approach to q-binomial theorem

3. Proof of the n-space q-binomial theorem.

4. Finite Product 2D, 3D and nD cases of q-binomial extensions.

Chapter 14. Weighted Vector Partitions as hybrid n-space variations

1. Examples of hybrid variations for identities generating weighted vector

partitions

2. Proof of the n-space hybrid q-binomial theorem.

Chapter 15. Functional Equations for n-space Vector Partitions

1. An n-space q-binomial functional equation.

2. An n-space binary vector partitions functional equation.

Chapter 16. Binary Partitions and their Vector Generalizations

1. Introductory Elementary Ideas for Binary Partitions.

2. A few finite and infinite products for binary partitions.

3. 2D version of every integer is a unique sum of distinct binary powers

4. The 2D binary, n-ary and 10-ary formulas

5. Some binary integer partition preliminary results.

6. Some easy 2D binary partition transform generating functions.

7. A binary partition 2 -space variation of extended q-binomial theorem.

8. First quadrant 2D binary partitions

9. First quadrant lower diagonal 2D binary partitions

10. First hyperquadrant 3D binary partitions

Chapter 17. n-ary Partitions and their Vector Generalizations

1. Integer n-ary partitions.

2. A Base 10 or 10-ary set of cases

3. The n-ary integer partition set of cases

Chapter 18. Some Binary and n-ary Partitions Analytic Formulas

1. Historical remarks for binary partitions analytic formulas

2. Some lacunary series and products for n-ary partitions

3. Binary and n-ary versions of Products for Distinct Partitions

Chapter 19. Visible Point Vector Identities in the first Hyperquadrant

1. Introducing the VPV identities.

2. Deriving the 2D first quadrant VPV identity.

3. Deriving the n-dimensional first hyperquadrant VPV identity.

4. Diversionary note on the abc Conjecture.

5. Application of a VPV identity to the abc Conjecture.

Chapter 20. VPV Identity cases related to x^{y} = y^{x} and x^{y}y^{x} = v^{w}w^{v}.

1. Rational solutions of x^{y} = y^{x}

2. Solutions to x^{y}y^{x} = v^{w}w^{v} in rationals and integers

3. VPV Identity transforms using x^{y} = y^{x} and x^{y}y^{x} = v^{w}w^{v}.

Chapter 21. Visible Point Vector Identities in Hyperpyramid lattices

1. VPV identities in square hyperpyramid regions.

2. Deriving 2D VPV identities in square hyperpyramid regions.

3. Deriving 3D VPV identities in square hyperpyramid regions.

4. VPV identities in nD square hyperpyramid regions.

Chapter 22. Polylogarithms, and Parametric Euler Sum Identities.

1. Early history of the Dilogarithm

2. The Trilogarithm function

3. The Polylogarithm function

4. Mordell-Tornheim-Witten ensembles

5. Finite Euler Sums

5.1. The 2D square hyperpyramid VPV identity.

5.2. The 3D square hyperpyramid VPV identity.

6. Parametric Euler Sum Identities

Chapter 23. Visible Point Vector identities related to particular Euler sum values

1. Polylogarithms near trivial zeroes of the Riemann zeta function

2. Identities near non-trivial zeroes of the Riemann zeta function

Chapter 24. Visible Point Vector Identities in Skewed Hyperpyramid lattices

1. Visible Point Vector identities in asymmetric hyperpyramid regions.

1.1. A more general hyperpyramid theorem.

Chapter 25. The Ramanujan trigonometric function and visible point identities

1. Introductory remarks

2. Dirichlet series generating functions

3. Proof of Theorem 2.1

4. A new Jordan Totient generating function, and some related results

5. Further multidimensional formulae

6. Application of Jacobi theta series to the generalized summations.

Chapter 26. Other Non-weighted n-space Vector Partition Theorems

1. A binary partition 2 -space variation of extended q-binomial theorem.

Chapter 27. Determinants, Bell Polynomial Expansions for Vector Partitions

1. Some reference cases of the determinants in this book.

2. Calculus of Determinant Evaluation

Chapter 28. The 2D and 3D Weighted Stepping Stone Models

1. Introduction

2. Heuristic concepts for the known results

Chapter 29. The 2D and 3D Light Diffusion Models

1. The 2D Light Diffusion Model in the first quadrant.

2. Partition Grids for 2D Vector partitions

3. Partition Grids for unweighted 2D VPVs

Chapter 30. Partition Grids for unweighted 2D VPVs II

Chapter 31. Partition Grids for unweighted 2D VPVs III

Chapter 32. Partition Grids for weighted 2D VPVs IV

Chapter 33. The 3D Light Diffusion Model in the first hyperquadrant.

Bibliography

**Other projects I have:**

In 2006 I published an introductory paper for ** Dirichlet series analogues of q-series**, which led me to arithmetical function identities encoding so-named

**quasicrystals**. I give new analogue summations for classical

*q*-series and hypergeometric series summations in terms of

*Riemann Zeta functions*and

*Jordan Totient functions*. This area of research is still being developed, and I am drafting a monograph on this topic as well, all of it sequel to the 2006 paper.

I therefore can say I have research interests in the Theory of Higher Dimensional Partitions, Aperiodic Order, Dynamical Systems, Combinatorics, Discrete Geometry, Number Theory, Quasicrystal tilings and their Dirichlet series functions, and Mathematical Physics, and areas where these theories may overlap.

I am also Manager of the almost 40,000 member ** LinkedIn Number Theory Group** located at https://www.linkedin.com/groups/4510047/. I have posted many brief mathematical problems and news stories at graduate and research levels over the past seven years in that forum. An informative post in that group is http://bit.ly/3a6Wyhc.

I am also an Administrator for the Facebook Group ** Classical Mathematics** presently with about 20,000 members. A typical post in that group is http://bit.ly/2NFtbLn.

### Groups

- Collaborator, Mathematical physics

**I am a published poet as well as a mathematics person. **

**POEMS: **

In 2010 I published *Words in Common*, which is a collection over decades of *journal/magazine/anthology* published poems partly funded by the Australia Council for the Arts many years ago. The book was officially launched by Professor Kevin Brophy from University of Melbourne, Creative Writing Department. It was edited by Associate Professor Trevor Code from Deakin University.

See a link with excerpts from the poems at http://bit.ly/3jAvlb8.

See a review of *Words In Common* by Canberra poet Michael Byrne at http://bit.ly/2MRlv8y.

**A SELECTION OF MY MATHEMATICAL PAPERS:**

31 CAMPBELL, G. B. *Fun with numbers: Rational solutions to x ^{y}y^{x} = v^{w}w^{v}*

*,*Aust. Math. Soc. Gazette, Volume 49, No5, To appear November 2022. (https://austms.org.au/publications/gazette/gazette495/)

30 CAMPBELL, G. B. *Fun with numbers: Identities containing a certain algebraic form**, * Aust. Math. Soc. Gazette, Volume 49, No4, pp162-163, September 2022. (https://austms.org.au/publications/gazette/gazette494/)

29 CAMPBELL, G. B. *Fun with numbers: Fun with numbers: Consecutive 6th powers and base 6 numbers, * Aust. Math. Soc. Gazette, Volume 49, No3, pp108-109, July 2022. (https://austms.org.au/publications/gazette/gazette493/)

28 CAMPBELL, G. B. *Fun with numbers: Ramanujan 6-10-8 identity, * Aust. Math. Soc. Gazette, Volume 49, No2, pp71-72, May 2022. (https://austms.org.au/publications/gazette/gazette492/)

27 CAMPBELL, G. B. *An interview with Rodney James Baxter*, Aust. Math. Soc. Gazette, Volume 47, No1, pp24-32, March 2020. (https://austms.org.au/wp-content/uploads/2020/07/471Web.pdf)

26. CAMPBELL, G. B. Some n-space q-binomial theorem extensions and similar identities, arXiv:1906.07526v1 [math.NT], Jun 2019. (https://arxiv.org/abs/1906.07526)

25 CAMPBELL, G. B. and ZUJEV, A. The series that Ramanujan misunderstood, arXiv:1610.03693v1 [math.NT], Oct 2016. (https://arxiv.org/abs/1610.03693v1)

24 CAMPBELL, G. B. and ZUJEV, A. On integer solutions to x^{5} - (x+1)^{5} - (x+2)^{5} + (x+3)^{5} = 5^{m} + 5^{n}, arXiv:1603.00080v1 [math.NT], Feb 2016. (https://arxiv.org/abs/1603.00080v1)

23 CAMPBELL, G. B. and ZUJEV, A. Some equations with features of digit reversal and powers, arXiv:1602.06320v1 [math.NT], Feb 2016. (https://arxiv.org/abs/1602.06320v1)

22 CAMPBELL, G. B. and ZUJEV, A. Gaussian integer solutions for the fifth power taxicab number problem, arXiv:1511.07424v1 [math.NT], Nov 2015. (https://arxiv.org/abs/1511.07424v1)

21 CAMPBELL, G. B. and ZUJEV, A. Variations on Ramanujan's nested radicals, arXiv:1511.06865v1 [math.NT], Nov 2015. (https://arxiv.org/abs/1511.06865v1)

20 CAMPBELL, G. B. and ZUJEV, A. A diophantine sum with factorials, arXiv:1510.03056v2 [math.NT], Oct 2015. (https://arxiv.org/abs/1510.03056v2)

19 CAMPBELL, G. B. The q-Dixon sum Dirichlet series analogue, arXiv:1302.2664v1, Feb 2013. (https://arxiv.org/abs/1302.2664v1)

18 CAMPBELL, G. B. Ramanujan and Eckford Cohen totients from Visible Point Identities, arXiv:1212.2818v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2818v1)

17 CAMPBELL, G. B. D-analogues of q-shifted factorial and the q-Kummer sum, arXiv:1212.2248v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2248v1)

16 CAMPBELL, G. B. Polylogarithm approaches to Riemann Zeta function zeroes, arXiv:1212.2246v1 [math.NT], Dec 2012. (https://arxiv.org/abs/1212.2246v1)

15 CAMPBELL, G. B. Dirichlet series analogues of q-shifted factorial and the q-Kummer sum, Research paper 2003-6, Department of Mathematics, LaTrobe University, 2003.

14 CAMPBELL, G. B. An Euler Product transform applied to q series, Ramanujan J (2006) 12:267-293. (https://doi.org/10.1007/s11139-006-0078-y)

13 CAMPBELL, G. B. A New Class of Identities akin to q-Series in Several Variables, Research paper no (to be determined), Centre for Mathematics and its applications, The Australian National University, 1998.

12 CAMPBELL, G. B. Combinatorial Identities in Number Theory related to q-series and Arithmetical functions, Bull. Austral. Math. Soc., Vol. 58, (1998) pp345-347.

11 CAMPBELL, G. B. On generating functions for vector partitions, Research paper no 55-97, Centre for Mathematics and its applications, The Australian National University, 1997.

10 CAMPBELL, G. B. Visible point vector summations from hypercube and hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 21, No 4, 741-748, 1998. (https://www.researchgate.net/publication/26536267_Visible_point_vector_summations_from_hypercube_and_hyperpyramid_lattices)

9 CAMPBELL, G. B. Infinite products over hyperpyramid lattices, Internat. J. Math. & Math. Sci., Vol 23, No 4, 2000, 271-277. (http://downloads.hindawi.com/journals/ijmms/2000/108918.pdf)

8 CAMPBELL, G. B. A closer look at some new identities, Internat. J. Math. & Math. Sci., Vol 21, No 3, 1998, pp581-586. (https://www.researchgate.net/publication/26536244_A_closer_look_at_some_new_identities)

7 CAMPBELL, G. B. Infinite products over visible lattice points, Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 637-654. (http://downloads.hindawi.com/journals/ijmms/1994/705467.pdf)

6 CAMPBELL, G. B. A new class of infinite product, and Euler's totient, Internat. J. Math. & Math. Sci., Vol 17, No 4, 1994, 417-422.

5 CAMPBELL, G. B. Formulae with functions exhibiting self-similarity, Research Paper preprint series, Centre for Mathematics and its Applications, The Australian National University, 1993.

4 CAMPBELL, G. B. A generalised formula of Hardy, Int. J. Math. Math. Sci., Vol 17, No 2, 1994, 369-378.

3 CAMPBELL, G. B. Dirichlet summations and products over primes, Internat. J. Math. & Math. Sci., Vol 16, No 2, 1993, 359-372.

2 CAMPBELL, G. B. Multiplicative functions over Riemann zeta function products, J. Ramanujan Soc. 7 No. 1, 1992, 52-63.

1 CAMPBELL, G. B. Generalization of a formula of Hardy, La Trobe University preprints no 79-5, 1979 (written whilst a young student.)