James McLaughlin

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My Research Statement (years out of date, and these days I am more interested in basic hypergeometric series and related areas such as integer partitions, but I am leaving it here since I have nothing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.

Students can access course information through D2L.

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 - Ann. Comb. 27, 329–352 (2023). Annals of Combinatorics. PDF
55. 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
56. 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
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) - New York J. Math. 30 (2024) 1307–1436. PDF
59. Dissections of lacunary eta quotients and identically vanishing coefficients  (with Tim Huber and Dongxi Ye) - Submitted.  PDF
60. 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).
61. Asymptotics and Sequential Closures of Continued Fractions and Generalizations (with Douglas Bowman)
    - submitted. PDF

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