James McLaughlin
| Office: | UNA 108, 25 University Avenue |
| Phone: | 610-430-4417 |
| Fax: | 610-738-0578 |
| Email: | James McLaughlin |
Address:
Department of Mathematics
Room 101, 25 University Avenue
West Chester University
West Chester, PA 19383
My Curriculum Vitae: PDF
This presentation tries to put together a glimpse of the picture of the Cosmos, and the relationship of humans to the Cosmos, the overall "big picture" of Existence, so far as we understand it, that our education system should be passing on to future generations of humans, if we are to successfully move past our present "grunting hooligan" stage, to become fully fledged Cosmic Citizens.
Talk Abstract Talk Presentation Talk Video
Talk 1 Presentation Talk 2 Presentation
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
If we replace the binomial power \((1+x)^n \) by the \(q\)-binomial power, \((1+x)(1+xq)\cdots(1+xq^{n-1})\), and expand it as a polynomial in \(x\), we get a generalization of the binomial coefficient in which \( (1-q)(1-q^2)\cdots(1-q^n)/(1-q)^n\)plays the role of \(n!\). From this simple observation, \(q\)-series emerge.
Over the past half century, the world of \(q\)-series has grown to become one of the most exciting areas of mathematical research. Its origins lie in Euler's investigations of the generating functions for partitions and Jacobi's theta functions. It has proven to be intimately tied to many combinatorial problems from tilings to knot theory as well as to an understanding of special functions and orthogonal polynomials, Lie algebras and representation theory, and aspects of mathematical physics that range from statistical mechanics to particle theory. Explorations of its hidden patterns have spurred the ongoing development of computer algebra.
The relevance of \(q\)-series to contemporary mathematics truly blossomed in the 1970's due to a confluence of factors that included I. G. Macdonald's realization of the connection between affine root systems and the Dedekind \(\eta\)-function [Invent. Math. 15 (1972), 91–143; MR0357528], the start of a long and productive collaboration between G. E. Askey and R. A. Andrews exploring the \(q\)-analogs of orthogonal polynomials [see especially G. E. Askey and R. A. Andrews, in Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976), 3–26, NATO Adv. Study Inst. Ser., Ser. C: Math. Phys. Sci., 31, Reidel, Dordrecht, 1977; MR0519776], the publication of Andrews' foundational work The theory of partitions [reprint of the 1976 original, Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1998; MR1634067], B. C. Berndt's decision to systemically explain and—where possible—prove every identity in Ramanujan's personal notebooks (the last of these volumes is [Ramanujan's notebooks. Part V, Springer, New York, 1998; MR1486573]), Andrews' discovery of Ramanujan's "Lost Notebook'' on mock theta functions [Amer. Math. Monthly 86 (1979), no. 2, 89–108; MR0520571], and R. J. Baxter's recognition of the role of \(q\)-series in statistical mechanics [Exactly solved models in statistical mechanics, Academic Press, London, 1982; MR0690578].
No single book could possibly serve as an introduction to the breadth of this field
today. Mc Laughlin, in the book under review, has undertaken a simpler but still ambitious
goal: to introduce the reader to the world of basic hypergeometric series and their
identities. This is not the first introduction to these series. W. N. Bailey included
a chapter on them in his 1935 Generalized hypergeometric series [reprint, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner,
Inc., New York, 1964; MR0185155]. In 1990, G. Gasper Jr. and M. Rahman published what
is—now in its second edition—the standard encyclopedic reference to these series [Basic hypergeometric series, second edition, Encyclopedia Math. Appl., 96, Cambridge Univ. Press, Cambridge,
2004; MR2128719]. But Gasper and Rahman's book is both denser and focused on \(q\)-orthogonal
polynomials.
Mc Laughlin makes very different but equally valid choices of topics. The early chapters
come in bite sizes with lots of exercises: the \(q\)-binomial theorem, fundamental
transformation formulæ, Jacobi's triple product identity, Ramanujan's \(_1\psi_1\)
summation, Bailey's \(_6\psi_6\) summation, and the Rogers-Fine identities. Mc Laughlin
then moves into the heart of the book, Bailey pairs and Bailey chains, one of the
essential tools for the exploration of \(q\)-series. The text finishes by tackling
a selection of distinct topics shaped largely by the author's own interests, including
\(q\)-trinomial coefficients, a short chapter on partitions and bijective proofs,
an introduction to Lambert series, and a brief survey of mock theta functions.
There is a more extensive treatment of Mc Laughlin's first love, continued fractions.
While, as Mc Laughlin explains in his introduction, it was Ramanujan's continued fraction
$$ K(q) :=1+\frac{q}{1+\frac{q^2}{1+\frac{q^3}{1+\frac{q^4}{\ddots}}}} $$
that first drew him to the study of \(q\)-series, he never gets to the punchline,
that Ramanujan used his product representation of this continued fraction to prove
that
$$ \frac{1}{K(e^{-2\pi})} = e^{2\pi/5}\left(\sqrt{\frac{5+\sqrt{5}}{2}} - \frac{\sqrt{5}+1}{2}\right),
$$
one of two evaluations of \(K\) at special values that convinced Hardy that, even
though he had no idea how to prove them, "they must be true because, if they were
not true, no one would have had the imagination to invent them.''
Of course, this evaluation relies on the theory of theta functions. A work such as this, intended as an introductory textbook, is necessarily circumscribed in its reach. But it does serve as a reminder that beyond this book lies a vast expanse of \(q\)-series that has barely been touched.
Mc Laughlin concludes with seven appendices, intended to serve as a reference. These complement the work of Gasper and Rahman by providing extensive lists of Bailey pairs and Bailey chains. 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.
(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:
- 2019 Joint Meetings in Baltimore (co-organized with Geremías Polanco and Barry Smith)
- 2017 Joint Meetings in Atlanta (co-organized with Geremías Polanco)
- 2015 Joint Meetings in San Antonio
- 2013 Joint Meetings in San Diego
- 2011 Joint Meetings in New Orleans
- 2009 Joint Meetings in Washington, D.C.
- 2006 Joint Meetings in San Antonio
- 2004 Joint Meetings in Phoenix
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.
Teaching
| 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.
Book
Publications/Preprints
Note that the preprint version of papers on this page will differ to some extent from the version which eventually appeared in print.
- Polynomial Continued Fractions (With D. Bowman)
- Acta Arith. 103 (2002), no. 4, 329--342. PDF - 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 - 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 - 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 - 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 - 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 - Real Numbers with Polynomial Continued Fraction Expansions (with Nancy Wyshinski)
- Acta Arith. 116 (2005), no. 1, 63-79. PDF - 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 - 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 - 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 - Powers of a matrix and combinatorial identities (with B. Sury)
- INTEGERS: The Electronic Journal of Combinatorial Number Theory 5 (2005), A13, 9 pp. PDF - 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 - The Convergence behavior of $q$-Continued Fractions on the Unit Circle (With D. Bowman)
- The Ramanujan Journal 12 (2006), no. 2, 185--195. PDF - Ramanujan and Extensions and Contractions of Continued Fractions (with Nancy Wyshinski)
- The Ramanujan Journal 14 (2007), no. 3, 389--404. PDF - 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. - Continued Fractions with Multiple Limits (With D. Bowman)
- Advances In Mathematics Volume 210 (2007), no. 2, 578-606. PDF - A $q$-continued fraction (with Doug Bowman and Nancy Wyshinski)
- International Journal of Number Theory Volume 2 (2006), no. 4, 523-547. PDF - 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 - 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 - Some more Long Continued Fractions, I (with Peter Zimmer)
- Acta Arith.127 (2007), no. 4, 365--389. PDF - Symmetry and specializability in the continued fraction expansions of some infinite
products
- The Journal of Number Theory 127 (2007), no. 2, 184--219. PDF - 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 - Rogers-Ramanujan-Slater Type Identities (with Andrew Sills and Peter Zimmer)
- Electronic Journal of Combinatorics 15 (2008) #DS15, 59 pp. PDF - 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 - 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 - 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 - Some new Families of Tasoevian- and Hurwitzian Continued Fractions
- Acta Arith.. 135 (2008), no. 3, 247--268. PDF - Some Implications of the WP-Bailey Tree (with Peter Zimmer)
- Advances in Applied Mathematics Volume 43, Issue 2, August 2009, Pages 162-175 . PDF - 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 - Rogers-Ramanujan Computer Searches (with Andrew Sills and Peter Zimmer)
- Journal of Symbolic Computation Volume 44, Issue 8, August 2009, Pages 1068-1078 PDF - Lifting Bailey Pairs to WP-Bailey Pairs (with Andrew Sills and Peter Zimmer)
- Discrete Mathematics 309 (2009), pp. 5077-5091 PDF - An Identity Motivated by an Amazing Identity of Ramanujan
- The Fibonacci Quarterly 48 (2010), no. 1, 34–38, . PDF - General WP-Bailey Chains (with Peter Zimmer)
- The Ramanujan Journal 22 (2010), no. 1, 11–31. PDF - Some new Transformations for Bailey pairs and WP-Bailey Pairs - Central European Journal of Mathematics 8 (2010), no. 3, 474–487. PDF
- 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
- Some Applications of a Bailey-type Transformation (with Peter Zimmer)
- The International Mathematical Forum Vol. 5, 2010, no. 61-64, 3007-3022. PDF - A New Summation Formula for WP-Bailey Pairs - Applicable Analysis and Discrete Mathematics (AADM) 5 (2011), 67-79 PDF
- 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
- 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
- Polynomial Generalizations of two-variable Ramanujan type identities (with Andrew Sills) - Electronic Journal of Combinatorics Volume 18(2), 2011, P15 PDF
- 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 - On a pair of identities from Ramanujan's lost notebook (with Andrew Sills) - Annals of Combinatorics Volume 16, Number 3 (2012), 591-607 PDF
- A Reciprocity Relation for WP-Bailey Pairs (with Peter Zimmer) - The Ramanujan Journal Volume 28, Number 2 (2012), 155-173 PDF
- Further results on vanishing coefficients in infinite product expansions - J. Aust. Math. Soc. 98 (2015), no. 1, 69–77. PDF
- General multi-sum transformations and some implications - The Ramanujan Journal April 2016, Volume 39, Issue 3, pp 545-565 PDF
- Refinements of Some Partition Inequalities
- INTEGERS: The Electronic Journal of Combinatorial Number Theory 16 (2016), A66, 11 pp. PDF - 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 - 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 - 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 - 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 - 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
- m-Dissections of some infinite products and related identities - The Ramanujan Journal 59, pages313–350 (2022) PDF
- 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
- Some more identities of Kanade--Russell type derived using Rosengren's method - Ann. Comb. 27, 329–352 (2023). Annals of Combinatorics. PDF
- Lacunary eta quotients with identically vanishing coefficients (with Tim Huber and Dongxi Ye) - Vol. 19, No. 07, pp. 1639-1670 (2023) International Journal of Number Theory. PDF
- On the vanishing of the coefficients of CM eta quotients (with Timothy Huber, Chang Liu, Dongxi Ye, Maiodan Yuan and Sumeng Zhang) - Volume 66 , Issue 4 , November 2023 , pp. 1202 - 1216 Proceedings of the Edinburgh Mathematical Society. PDF
- 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
- Further Results on Vanishing Coefficients in the Series Expansion of Lacunary Eta Quotients (with Tim Huber and Dongxi Ye) - New York J. Math. 30 (2024) 1307–1436. PDF
- Dissections of lacunary eta quotients and identically vanishing coefficients (with Tim Huber and Dongxi Ye) - Submitted. PDF
- Identical Vanishing of Coefficients in the Series Expansion of Eta Quotients, modulo 4, 9 and 25 (with Tim Huber and Dongxi Ye) - Submitted PDF ( link to tables).
- Asymptotics and Sequential Closures of Continued Fractions and Generalizations (with
Douglas Bowman)
- submitted. PDF
Mathematical Ancestry
- Douglas Bowman (Thesis Adviser)
- Basil Gordon
- Tom M. Apostol
- Jacob David Tamarkin
- Andrei Andreevich Markov
- Pafnuty Lvovich Chebyshev
- Nikolai Dmitrievich Brashman
- J. J. Von Littrov