James McLaughlin

James McLaughlin
Office: UNA 108, 25 University Avenue
Phone: 610-430-4417
Fax: 610-738-0578
Email: James McLaughlin

Department of Mathematics
Room 101, 25 University Avenue
West Chester University
West Chester, PA 19383

My Curriculum Vitae: PDF

Talk Abstract                                           Talk Presentation                                    Talk Video

Talk 1 Flyer                                                  Talk 2 Flyer

Talk 1 Presentation                               Talk 2 Presentation     

Talk 1 Video                                             Talk 2 Video

Here is a video of a talk I gave via Zoom at the West Coast Number Theory Conference on 12/17/2022.

Issue #2 of the Mathematics Department's magazine, Math Times , has been published. Yours truly did most of the editing and compiling of things together. A big thank you to everyone who contributed items. Enjoy!

"The Trustees Achievement Award recognizes distinguished scholarly and creative work by a current West Chester University faculty member: specific, original contributions to a discipline (as in a research study, monograph, scholarly paper, literary work, musical composition, or work of art), or a significant collective body of work. Emphasis is on the national and international contribution of the scholarly work."

I had held off on posting a link to the review of my book, Topics and Methods in q-Series, by David M. Bressoud on the AMS MathSciNet website. MathSciNet needs a subscription and thus the review would not have been freely available. However, Howard Cohl at NIST recently informed me that he had received permission to publish the full review in the latest OPSF newsletter, so I obtained the same permission*. Here is the

full review

(it may be necessary to hard refresh the page (CTRL + F5) to get Mathjax to work in order to properly view the math equations), and here is the concluding sentence from the review:

"Mc Laughlin has produced an admirable book, clearly and knowledgeably written, upon which one could build a challenging undergraduate seminar as well as a graduate course designed to lead toward today's research questions."

*This work was originally published by the American Mathematical Society (AMS) as the review MR3752164 in Mathematical Reviews / MathSciNet. See https://mathscinet.ams.org/mathscinet-getitem?mr=3752164. It is reprinted here by permission of the AMS.

New: MAA review of Topic and methods in q-series I just became aware that my book, Topics and Methods in q-Series, was given a quite favourable review by Michael Berg, Professor of Mathematics at Loyola Marymount University in Los Angeles, CA, in MAA reviews on 05/23/2018.

"In closing, let me also note that Topics and Methods in q-Series, in addition to presenting such a wonderful sweep of deep and beautiful material, is very strong pedagogically. It’s very well written, in an accessible and clear style, the material dealt with is effectively motivated and discussed in a sound and rigorous manner, all the proofs are there, and McLaughlin gives the reader a large number of exercises to do along the way, as he travels these paths. Again, the expressions and formulas that pepper the pages of this book are at first sight daunting, but they do yield to persistent pressure, as per Andrews’ observation cited above. The language of q-series makes for some beautiful poetry."

New: Cellular Automaton While preparing some material for my MAT 413 Computer Algebra class, I discovered (or maybe rediscovered) a pair of quite interesting (at lease to me) cellular automata. If anyone has seen them before and can direct me to any information online, or published articles, about them, I would very much appreciate it. If you follow the link to the page I created about them, you will also find some open math questions about them.

Update (12/10/2018): I was today informed that the first of these cellular automata is well-known and extensively studied. A google search for "B1357/S1357", one of its official classifications, will bring up many hits. It is also known as "Replicator", and was discovered by Edward Fredkin.

The inaugaral issues the Mathematics Department's newsletter, Math Times , has just been published. Yours truly created most of the layout and "stitched" everything together. Its a little crude for a first attempt, but issue 2 will hopefully look a little more polished.

My (not so new) views on what should be included in school curricula.

Geremías Polanco, Barry Smith, Nancy Wyshinski and I are organizing a special session on continued fractions at the 2019 Joint Meetings in Baltimore.

My book, Topics and Methods in q-Series, appeared in print in November 2017. It is being published by World Scientific. It is available for purchase on Amazon and elsewhere.

Nancy Wyshinski and I also organized previous special session on continued fractions at the following meetings:

Pictures from the special session in San Antonio, and abstracts of the talks.
Schedule of the Phoenix special session.

Research Interests

My present interests are in the area of basic hypergeometric series and related areas, such as integer partitions.

I am currently also working on various problems related to continued fractions.

I have also investigated various convergence problems for q-continued fractions, and I and my thesis adviser, Douglas Bowman, recently partially settled a long-standing open problem on the convergence of the Rogers-Ramanujan continued fraction on the unit circle.

I am also interested in the problem of finding the regular continued fraction expansion of a number expressed in some other form (for example, as an infinite series or infinite product) and finding numbers with predictable patterns in their regular continued fraction expansions.

Another area of interest is the evaluation of polynomial continued fractions.

I am also interested in various problems in the area of Diophantine equations.

My Research Statement (years out of date, and these days I am more interested in basic hypergeometric series and related areas such as integer partitions, but I am leaving it here since I have nothing more current to replace it with): DVI PS PDF

My publications list on MathSciNet.


Fall 2017 Teaching Schedule:
Course # Course Name
MAT 151-01 (6238) Intro Discrete Math (Lecture)
MAT 151-02 (1984) Intro Discrete Math (Lecture)
MAT 411-01 (2018) Undergraduate Algebra I (Lecture)
MAT 515-01 (8082) Graduate Algebra I (Lecture)

Students can access course information through D2L.


My first book, Topics and Methods in q-Series, will be published by World Scientific, in their Monographs in Number Theory series.


Note that the preprint version of papers on this page will differ to some extent from the version which eventually appeared in print.

  1. Polynomial Continued Fractions (With D. Bowman)
    - Acta Arith. 103 (2002), no. 4, 329--342. PDF
  2. On the Divergence of the Rogers-Ramanujan Continued Fraction on the Unit Circle (With D. Bowman)
    - The Transactions of the American Mathematical Society 356 (2004), no. 8, 3325--3347. PDF
  3. Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields
    - J. London Math. Soc. (2) 67 (2003), no. 1, 16--28. PDF
  4. Multi-variable Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields
    - Pacific J. Math. 210 (2003), no. 2, 335--349. PDF
  5. On The Divergence in the General Sense of $q$-Continued Fractions on the Unit Circle (With D. Bowman)
    - Communications in the Analytic Theory of Continued Fractions 11(2003), 25--49. PDF
  6. A Theorem on Divergence in the General Sense for Continued Fractions (With D. Bowman)
    - The Journal of Computational and Applied Mathematics 172, (2004) no. 2, pp 363--373. PDF
  7. Real Numbers with Polynomial Continued Fraction Expansions (with Nancy Wyshinski)
    - Acta Arith. 116 (2005), no. 1, 63-79. PDF
  8. Combinatorial Identities Deriving from the n-th Power of a $2 \times 2$ Matrix
    - INTEGERS: The Electronic Journal of Combinatorial Number Theory 4 (2004), A19, 14 pp. PDF
  9. A Convergence Theorem for Continued Fractions of the Form $K_{n=1}^{\infty}a_{n}/1$ (with Nancy Wyshinski)
    - The Journal of Computational and Applied Mathematics, Volume 179 , Issues 1--2, 1 July 2005, Pages 255-262, containing the proceedings from the Conference on Orthogonal Functions and Related Topics, Roros, Norway, August 2003. PDF
  10. Ramanujan and the Regular Continued Fraction Expansion of Real Numbers (with Nancy Wyshinski)
    - The Mathematical Proceedings of the Cambridge Philosophical Society, Volume 138 - Issue 03 - May 2005, pp 367 - 381. PDF
  11. Powers of a matrix and combinatorial identities (with B. Sury)
    - INTEGERS: The Electronic Journal of Combinatorial Number Theory 5 (2005), A13, 9 pp. PDF
  12. The Convergence and Divergence of $q$-Continued Fractions outside the Unit Circle (With D. Bowman)
    - The Rocky Mountain Journal of Mathematics 36 (2006), no. 3, 799--809. PDF
  13. The Convergence behavior of $q$-Continued Fractions on the Unit Circle (With D. Bowman)
    - The Ramanujan Journal 12 (2006), no. 2, 185--195. PDF
  14. Ramanujan and Extensions and Contractions of Continued Fractions (with Nancy Wyshinski)
    - The Ramanujan Journal 14 (2007), no. 3, 389--404. PDF
  15. Further Combinatorial Identities Deriving from the n-th Power of a $2 \times 2$ Matrix (with Nancy Wyshinski)
    - Discrete Applied Mathematics 154 (2006), no. 8, 1301--1308.
  16. Continued Fractions with Multiple Limits (With D. Bowman)
    - Advances In Mathematics Volume 210 (2007), no. 2, 578-606. PDF
  17. A $q$-continued fraction (with Doug Bowman and Nancy Wyshinski)
    - International Journal of Number Theory Volume 2 (2006), no. 4, 523-547. PDF
  18. Continued Fractions and Generalizations with Many Limits: A Survey. (with Doug Bowman)
    - In the Proceedings of the Conference on Diophantine Analysis and Related Fields, Keio University, Yokohama, JAPAN, 7-10 March, 2006 , published as Seminar on Mathematical Sciences, 35, Keio University, Department of Mathematics, Yokohama (2006), 19-38. PDF
  19. Some properties of the distribution of the numbers of points on elliptic curves over a finite prime field (with Saiying He)
    - Bulletin of the Australian Mathematics Society 75 (2007), no. 1, 135--149. PDF
  20. Some more Long Continued Fractions, I (with Peter Zimmer)
    - Acta Arith.127 (2007), no. 4, 365--389. PDF
  21. Symmetry and specializability in the continued fraction expansions of some infinite products
    - The Journal of Number Theory 127 (2007), no. 2, 184--219. PDF
  22. Some Observations on Khovanskii's Matrix Methods for extracting Roots of Polynomials (with B. Sury)
    - INTEGERS: The Electronic Journal of Combinatorial Number Theory 7 (2007), A48, 12 pp. PDF
  23. Rogers-Ramanujan-Slater Type Identities (with Andrew Sills and Peter Zimmer)
    - Electronic Journal of Combinatorics 15 (2008) #DS15, 59 pp. PDF
  24. Ramanujan-Slater Type Identities Related to the Moduli 18 and 24 (with Andrew Sills)
    - The Journal of Mathematical Analysis and Applications 344/2 (2008) 765-777. PDF
  25. Some identities between basic hypergeometric series deriving from a new Bailey-type transformation (with Peter Zimmer)
    - The Journal of Mathematical Analysis and Applications , 345/2 (2008) 670-677. PDF
  26. Some more identities of the Rogers-Ramanujan type (with Douglas Bowman and Andrew Sills)
    - The Ramanujan Journal Volume 18, Issue 3 (2009), Page 307 PDF
  27. Some new Families of Tasoevian- and Hurwitzian Continued Fractions
    - Acta Arith.. 135 (2008), no. 3, 247--268. PDF
  28. Some Implications of the WP-Bailey Tree (with Peter Zimmer)
    - Advances in Applied Mathematics Volume 43, Issue 2, August 2009, Pages 162-175 . PDF
  29. Combinatorics of Ramanujan-Slater Type Identities (with Andrew Sills)
    - Combinatorial Number TheoryProceedings of the 'Integers Conference 2007', Carrollton, Georgia, USA, October 24—27, 2007, 125–139, Walter de Gruyter, Berlin, 2009. PDF
  30. Rogers-Ramanujan Computer Searches (with Andrew Sills and Peter Zimmer)
    - Journal of Symbolic Computation Volume 44, Issue 8, August 2009, Pages 1068-1078 PDF
  31. Lifting Bailey Pairs to WP-Bailey Pairs (with Andrew Sills and Peter Zimmer)
    - Discrete Mathematics 309 (2009), pp. 5077-5091 PDF
  32. An Identity Motivated by an Amazing Identity of Ramanujan
    - The Fibonacci Quarterly 48 (2010), no. 1, 34–38, . PDF
  33. General WP-Bailey Chains (with Peter Zimmer)
    - The Ramanujan Journal 22 (2010), no. 1, 11–31. PDF
  34. Some new Transformations for Bailey pairs and WP-Bailey Pairs - Central European Journal of Mathematics 8 (2010), no. 3, 474–487. PDF
  35. Continued Fraction Proofs of $m$-versions of Some Identities of Rogers-Ramanujan-Slater Type (with Doug Bowman and Nancy Wyshinski) - The Ramanujan Journal 25, Number 2, 203-227 PDF
  36. Some Applications of a Bailey-type Transformation (with Peter Zimmer)
    - The International Mathematical Forum Vol. 5, 2010, no. 61-64, 3007-3022. PDF
  37. A New Summation Formula for WP-Bailey Pairs - Applicable Analysis and Discrete Mathematics (AADM) 5 (2011), 67-79 PDF
  38. Some implications of Chu's $_{10}\psi_{10}$ extension of Bailey's $_{6}\psi_{6}$ summation formula (with Andrew Sills and Peter Zimmer) - Online Journal of Analytic Combinatorics (OJAC) Issue 5, 2010 PDF
  39. Hybrid Proofs of the $q$-Binomial Theorem and other identities (with Dennis Eichhorn and Andrew Sills) - Electronic Journal of Combinatorics Volume 18(1), 2011, P60 PDF
  40. Polynomial Generalizations of two-variable Ramanujan type identities (with Andrew Sills) - Electronic Journal of Combinatorics Volume 18(2), 2011, P15 PDF
  41. A Hardy-Ramanujan-Rademacher-type formula for $(r,s)$-regular partitions (with Scott Parsell)
    - The Ramanujan Journal June 2012, Volume 28, Issue 2, pp 253-271 PDF
  42. On a pair of identities from Ramanujan's lost notebook (with Andrew Sills) - Annals of Combinatorics Volume 16, Number 3 (2012), 591-607 PDF
  43. A Reciprocity Relation for WP-Bailey Pairs (with Peter Zimmer) - The Ramanujan Journal Volume 28, Number 2 (2012), 155-173 PDF
  44. Further results on vanishing coefficients in infinite product expansions - J. Aust. Math. Soc. 98 (2015), no. 1, 69–77. PDF
  45. General multi-sum transformations and some implications - The Ramanujan Journal April 2016, Volume 39, Issue 3, pp 545-565 PDF
  46. Refinements of Some Partition Inequalities
    - INTEGERS: The Electronic Journal of Combinatorial Number Theory 16 (2016), A66, 11 pp. PDF
  47. Applications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions (Jongsil Lee and Jaebum Sohn)
    - The Journal of Mathematical Analysis and Applications Volume 447, Issue 2, 15 March 2017, Pages 1126–1141. PDF
  48. Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series
    - Analytic number theory, modular forms and q-hypergeometric series, 503–531, Springer Proc. Math. Stat., 221, Springer, Cham, 2017. (This volume contains the refereed conference proceedings of the 2016 Gainesville International Number Theory Conference in honor of Krishna Alladi). PDF
  49. A Generalization of Schröter's Formula
    - Annals of Combinatorics, November 2019, Volume 23, Issue 3–4, pp 889–906, this volume comprising the refereed conference proceedings of Combinatory Analysis 2018, A Conference in Honor of George Andrews' 80th Birthday. PDF
  50. Some Observations on Lambert series, vanishing coefficients and dissections of infinite products and series
    - to appear in the book form of the refereed conference proceedings of Analytic and Combinatorial Number Theory: The Legacy of Ramanujan, A Conference In Honor Of Bruce C. Berndt's 80th Birthday. PDF
  51. New infinite q-product expansions with vanishing coefficients - The Ramanujan Journal 55 (2021), no. 2, 733–760. You may be able to access it via this Springer link. If not, here is link to a preprint version: PDF
  52. m-Dissections of some infinite products and related identities - The Ramanujan Journal 59, pages313–350 (2022) PDF
  53. Further results on Vanishing Coefficients in infinite products of the form \((q^b,q^{p-b};q^p)_{\infty}^3(q^{jb},q^{2p-jb};q^{2p})_{\infty}\) (with Peter Zimmer) - Vol. 18, No. 08, pp. 1863-1885 (2022) The International Journal of Number Theory. PDF
  54. Some more identities of Kanade--Russell type derived using Rosengren's method - published online May 12 2022 in Annals of Combinatorics. PDF
  55. Lacunary eta quotients with identically vanishing coefficients (with Tim Huber and Dongxi Ye) - published online (accepted January 11 2023) in International Journal of Number Theory. PDF
  56. On the vanishing of the coefficients of CM eta quotients (with Timothy Huber, Chang Liu, Dongxi Ye, Maiodan Yuan and Sumeng Zhang) - to appear in Proceedings of the Edinburgh Mathematical Society. PDF
  57. Generalizations of some q-product Identities of Ramanujan and others (with Timothy Huber and Dongxi Ye) - to appear in AMS Contemporary Mathematics Proceedings on hypergeometric functions, q-series and generalizations. PDF
  58. Further Results on Vanishing Coefficients in the Series Expansion of Lacunary Eta Quotients  (with Tim Huber and Dongxi Ye) - Submitted. PDF
  59. Dissections of lacunary eta quotients and identically vanishing coefficients  (with Tim Huber and Dongxi Ye) - Submitted.  PDF
  60. Asymptotics and Sequential Closures of Continued Fractions and Generalizations (with Douglas Bowman)
    - submitted. PDF

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