List of Publications

Most of the papers are available in DVI and PDF formats. Links to ABSTRACTS and reviews in MATHSCINET and ZENTRALBLATT are provided. The papers are listed in reverse chronological order.

Some papers contain SUPPLEMENTS. These supplements contain MAPLE programs for checking a paper and some additional notes.

Some papers have a PROOF version for proof-reading purposes.

[ArXiv] [PDF] [ABSTRACT]
F.G. Garvan and Rishabh Sarma, New symmetries for Dyson's rank function , Ramanujan J., to appear.
Original article: DOI
Article electronically published on April 3, 2024
(Posted August 18, 2023; updated April 7, 2024)
LOG

[ArXiv] [PDF] [ABSTRACT] [ArXiv] [PDF] [ABSTRACT]
Dandan Chen, Rong Chen and Frank Garvan, Congruences modulo powers of 5 for the rank parity function , Hardy-Ramanujan J., 45 (2020), 24-45.
Original article: DOI
Article electronically published on April 25, 2021
[MR 4298484]
(Posted January 5, 2021; updated April 7, 2024)
LOG

[ArXiv] [PDF] [ABSTRACT]
F.G. Garvan, A new approach to Dyson's rank conjectures , Ramanujan J., 61 (2023), 545-566.
FREE VIEW-ONLY VERSION
Original article: DOI
Article electronically published on February 5, 2022
(Posted December 15, 2020; updated May 16, 2023)
LOG

[ArXiv] [PDF] [ABSTRACT]
Rong Chen and F.G. Garvan, A proof of the mod 4 unimodal sequence conjectures and related mock theta functions, Advances in Mathematics 398 (2022), 108235.
FREE DOWNLOAD until April 8
Original article: DOI
Article electronically published on February 17, 2022
LOG

[PDF] [ABSTRACT]
Rong Chen and Frank Garvan, Congruences modulo 4 for weight $3/2$ eta-products , Bull. Austral. Math. Soc., 103 (2021), 405-417.
Original article: DOI
Article electronically published on 05 October 2020
(Posted Oct 5, 2020; Updated May 11, 2021)

[ArXiv] [ABSTRACT] [PDF] [SUPPLEMENTS]
Frank Garvan, A tutorial for the MAPLE ETA package, preprint (22 pages).
(Posted July 10, 2019)
LOG

[ABSTRACT]
Alexander Berkovich, Frank G. Garvan, and Hamza Yesilyurt, Ramanujan's circular summation, t-cores and twisted partition identities, J. Math. Anal. Appl., 479 (2019), no. 1, 773-788.
Original article: DOI
[MR 3987059]
(Posted June 13, 2019; updated August 20, 2019)
LOG

[ArXiv] [ABSTRACT] [PDF]
Frank Garvan, New fifth and seventh order mock theta function identities, Annals of Combinatorics, 23 (2019), no. 3-4, 765-783.
Original article: DOI
(Posted October 9, 2018, Updated December 8, 2019)
LOG

[PDF] [ABSTRACT] [SUPPLEMENTS]
Jie Frye and Frank Garvan, Automatic Proof of Theta-Function Identities , in "Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory", Texts Monogr. Symbol. Comput.,Springer, Cham,2019, pp.195-258.
[MR 3889559]
(Posted July 20, 2018; Updated August 20, 2019)
LOG

[ABSTRACT] [PREFACE]
Analytic Number Theory, Modular Forms and $q$-Hypergeometric Series; In Honor of Krishna Alladi's 60th Birthday, (Edited by G.E. Andrews and F. Garvan), Springer Proceedings in Mathematics & Statistics, Vol. 221, Springer, New York, 2018.
Original book: DOI
(Posted April 27, 2018)
LOG

[PDF] [ABSTRACT] [SUPPLEMENTS]
F. Garvan and M. J. Schlosser, Combinatorial interpretations of Ramanujan's tau function , Discrete Math. 341 (2018), no. 10, 2831--2840.
Original article: DOI
[MR 3843270]
(Posted June 26, 2016; Updated July 17, 2018)
LOG

[PDF] [ABSTRACT] [SUPPLEMENTS]
F.G. Garvan, Weighted partition identities and divisor sums , in "Frontiers in Orthogonal Polynomials and q-Series", Proceedings of a Conference in honor of Mourad Ismail for his 70th birthday, Orlando, May 10-12, 2015, World Scientific, pp.239-249.
Original article: DOI
[MR 3791622]
(Posted May 16, 2016; Updated May 20, 2016 and March 27, 2017)
LOG

[ArXiv] [PDF] [ABSTRACT]
F.G. Garvan, Transformation properties for Dyson's rank function , Trans. Amer. Math. Soc., 371 (2019), 199-248.
Original article: DOI
Article electronically published on May 30, 2018
[MR 3885143]
(Posted January 23, 2016; Updated June 17, 2016, February 23, 2017, July 20, 2018, and March 6, 2020)
LOG

[ArXiv] [PDF] [ABSTRACT]
F.G. Garvan and C. Jennings-Shaffer, Exotic Bailey-Slater SPT-functions II: Hecke-Rogers-type double sums and Bailey pairs From groups A, C, E , Adv. Math. 299 (2016), 605-639.
Original article: DOI
[MR 3519478]
[ZBL 1348.11079]
(Posted January 28, 2015; Updated August 4, 2016)
LOG

[ArXiv] [PDF] [ABSTRACT]
F.G. Garvan, Congruences and relations for r-Fishburn numbers , J. Combin. Theory Ser. A 134 (2015), 147-165.
Original article: DOI
[MR 3345301]
(Abstract posted May 13, 2014; PDF posted May 21, 2014; Revised June 20, 2014; Accepted March 13, 2015, Updated March 31,2015)
LOG

[ArXiv] [PDF] [ABSTRACT] [VIDEO]
F.G. Garvan, Universal mock theta functions and two-variable Hecke-Rogers identities , (Basil Gordon memorial issue) Ramanujan J. 36 (2015), no. 1-2, 267-296. Springer Nature View-Only:SharedIt
Original article: DOI
[MR 3296723 ]
(First posted December 24, 2013; PDF posted February 8, 2014; accepted August 1, 2014; updated August 19, 2014)
LOG

[ArXiv] [PDF] [ABSTRACT]
F.G. Garvan and C. Jennings-Shaffer, The spt-crank for overpartitions , Acta Arith., 166 (2014), no. 2, 141-188.
Original article: DOI
[MR 3277048]
(First posted November 14, 2013 to the arXiv; Accepted July 8, 2014)
LOG

[ArXiv] [PDF] [ABSTRACT]
F.G. Garvan and C. Jennings-Shaffer, Hecke-type congruences for Andrews' spt-function modulo 16 and 32 , Int. J. Number Theory, 10 (2014), no. 2, 375-390.
(First posted June 6, 2013 to the arXiv; Revised August 22, 2013; Updated September 14, 2013 and May 21, 2014)
[LOG]

[ArXiv] [PDF] [ABSTRACT]
F.G. Garvan and J.A. Sellers, Congruences for generalized Frobenius partitions with an arbitrarily large number of colors , INTEGERS, 14 (2014), Article A7.
(First posted May 26, 2013 to the arXiv, Updated April 21, 2014)
[LOG]

[ABSTRACT]
Computational and Analytical Mathematics: In Honor of Jonathan Borwein's 60th Birthday, Proceedings of the JonFest Workshop held at Simon Fraser University, Burnaby, BC, May 16-20, 2011, (Edited by D.H. Bailey, H.H. Bauschke, P. Borwein, F. Garvan, M. Thera, J.D. Vanderwerff, and Henry Wolkowicz), Springer Proceedings in Mathematics & Statistics, Vol. 50, Springer, New York, 2013.
Original article: DOI
[MR 3155260]
[LOG]

[ArXiv] [PDF] [ABSTRACT] [SUPPLEMENTS]
S.H. Chan, A. Dixit and F.G. Garvan, Rank-Crank type PDEs and generalized Lambert series identities , Ramanujan J. 31 (2013), no. 1-2, 163-189.
Original article: DOI
(First posted December 30, 2011; Updated February 23, 2012 and May 14, 2014)
[LOG]

[PDF] [ABSTRACT]
G.E. Andrews, F.G. Garvan and J.L. Liang, Self-conjugate vector partitions and the parity of the spt-function , Acta Arith., 158 (2013), no. 3, 199-218.
Original article: DOI
[MR 3040662]
LOG
(First posted November 14, 2011; Updated January 10, 2012; Revised December 11, 2012; Updated May 13, 2014)

[PDF] [ABSTRACT]
G.E. Andrews, F.G. Garvan and J.L. Liang, Combinatorial interpretations of congruences for the spt-function , (Ramanujan's 125th anniversary special volume) Ramanujan J. 29 (2012), no. 1-3, 321--338.
Original article: DOI
[MR 2994105]
LOG
(First posted October 12, 2011; Updated January 11, 2012, February 23, 2012 and December 12, 2012)

[ABSTRACT] [PREFACE]
Partitions, q-Series and Modular Forms, Proceedings of the conference held at the University of Florida, Gainesville, FL, March 2008 (Edited by Krishnaswami Alladi and Frank Garvan), Developments in Mathematics, 23, Springer, New York, 2012.
DOI
(First posted December 20, 2010)
[LOG]

[ArXiv] [DVI] [PDF] [ABSTRACT]
F.G. Garvan, Congruences for Andrews' spt-function modulo 32760 and extension of Atkin's Hecke-type partition congruences , Number theory and related fields, 165-185, Springer Proc. Math. Stat., 43, Springer, New York, 2013.

(Proceedings of an International Number Theory Conference in Memory of Alf van der Poorten, 12-16 March, 2012)
Original article: DOI
[MR 3081040]
[ZBL 06190321](First posted November 8, 2010; Updated August 8, 2012 and May 21, 2014)
LOG

[ArXiv] [DVI] [PDF] [ABSTRACT]
F.G. Garvan, Congruences for Andrews' spt-function modulo powers of 5, 7 and 13 , Trans. Amer. Math. Soc., 364 (2012), 4847-4873.
NOTE: Please send me any comments.
(First posted November 8, 2010; Updated December 14, 2010)

Original article: DOI
[MR 2922612]
[LOG]

[ArXiv] [PDF] [ABSTRACT]
F.G. Garvan, Higher order spt-functions, Adv. Math. 228 (2011), 241-265.
Original article: DOI
[MR 2822233]
(First posted August 6, 2010; Last updated May 25, 2011)
[LOG]
Original article: DOI

[ArXiv] [DVI] [PDF] [ABSTRACT]
F. G. Garvan, Biranks for partitions into 2 colors, in Ramanujan Rediscovered: Proceedings of a Conference on Elliptic Functions, Partitions, and q-Series in memory of K. Venkatachaliengar: Bangalore, 1-5 June, 2009, N. D. Baruah, B. C. Berndt, S. Cooper, T. Huber, and M. J. Schlosser, eds., Ramanujan Mathematical Society, Mysore, 2010, pp. 89-111.
(First posted September 26, 2009; Updated October 26, 2010)
[MR 2856959]
[LOG]

[ArXiv] [DVI] [PDF] [ABSTRACT]
Alexander Berkovich and Frank G. Garvan, The GBG-Rank and t-Cores I. Counting and 4-Cores, Journal of Combinatorics and Number Theory, 1 (2009), no. 3, 49--64.
[MR 2681308]
(Updated October 26, 2010)

[ArXiv]
K. Bringmann, F. G. Garvan, and K. Mahlburg, Partition statistics and quasiweak Maass forms, Int. Math. Res. Not. IMRN (2009), no. 1, 63-97.
Original article: DOI
[MR 2471296] [ZBL 1156.11021]
(Updated September 3, 2009)

[ArXiv] [DVI] [PDF] [ABSTRACT]
F.G. Garvan, Congruences for Andrews' Smallest Parts Partition Function and New Congruences for Dyson's Rank , Int. J. Number Theory, 6 (2010), no. 2, 1--29.
Original article: DOI
[MR 2646759]
(Updated March 6, 2010)

[ArXiv] [DVI] [PDF] [ABSTRACT]
Alexander Berkovich and Frank G. Garvan, K. Saito's Conjecture for Nonnegative Eta Products and Analogous Results for Other Infinite Products , J. Number Theory, 128 (2008), no. 6, 1731--1748.
Original article: DOI
[MR 2419190] [ZBL 1143.05008]

[ArXiv] [DVI] [PDF] [ABSTRACT]
Alexander Berkovich and Frank G. Garvan, The BG-rank of a partition and its applications, Adv. in Appl. Math. 40 (2008), no. 3, 377 - 400.
Original article: DOI
[MR 2402176]

[ArXiv] [DVI] [PDF] [ABSTRACT]
F.G. Garvan and Hamza Yesilyurt, Shifted and shiftless partition identities II, Int. J. Number Theory, 3 (2007), no. 1, 1--42.
Original article: DOI
[MR 2310493] [ZBL 1125.11057]

[ArXiv] [DVI] [PS] [PDF] [ABSTRACT]
Alexander Berkovich and Frank G. Garvan, On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo $5$ and generalizations, Trans. Amer. Math. Soc. 358 (2006), no. 2, 703--726.
Original article: DOI
[MR 2177037] [ZBL 1088.11073]

[ArXiv] [DVI] [PDF] [ABSTRACT]
Alexander Berkovich and Frank G. Garvan, Dissecting the Stanley partition function, J. Combin. Theory Ser. A 112 (2005), no. 2, 277--291.
Original article: DOI
[MR 2177487] [ZBL pre02235886]

[ArXiv] [DVI] [PDF] [ABSTRACT]
A.O.L. Atkin and F.G. Garvan, Relations between the ranks and cranks of partitions, (Rankin memorial issues) Ramanujan J. 7 (2003), no. 1-3, 343--366. Springer Nature View-Only:SharedIt
Original article available at www.springlink.com:
https://springerlink.metapress.com/openurl.asp?genre=article&id=doi:10.1023/A:1026219901284
[MR 2035811] [ZBL 1039.11069]
Warning: Published version has some typos. See [LOG]

[ArXiv] [DVI] [PDF] [ABSTRACT]
Alexander Berkovich and Frank G. Garvan, Some observations on Dyson's new symmetries of partitions, J. Combin. Theory Ser. A 100 (2002), no. 1, 61--93.
Original article: DOI
[MR 1932070] [ZBL 1016.05003]

[DVI] [PDF] [ABSTRACT]
F.G. Garvan, Shifted and shiftless partition identities, Number theory for the millennium, II (Urbana, IL, 2000), 75--92, A K Peters, Natick, MA, 2002.
[MR 1956245] [ZBL 1035.11050]

[DVI] [PDF] [ABSTRACT]
F.G. Garvan, More cranks and $t$-cores, Bull. Austral. Math. Soc. 63 (2001), no. 3, 379--391.
Original article: DOI
[MR 1834941] [ZBL 0985.11050]

[DVI] [PDF] [ABSTRACT]
Garvan, F. G.; A generalization of the Hirschhorn-Farkas-Kra septagonal numbers identity, Discrete Math. 232 (2001), no. 1-3, 113--118.
Original article: DOI
[MR 1823627] [ZBL 0979.05009]

[ABSTRACT]
Symbolic computation, number theory, special functions, physics and combinatorics, Proceedings of the conference held at the University of Florida, Gainesville, FL, November 11--13, 1999 (Edited by Frank G. Garvan and Mourad E. H. Ismail), Developments in Mathematics, 4, Kluwer Academic Publishers, Dordrecht, 2001.
[MR 1880075] [ZBL 0980.00026]

[ABSTRACT]
Frank Garvan, The Maple book, Chapman & Hall/CRC., Boca Raton, 2001.
[ZBL 1005.68185]

[DVI] [PDF] [MWS] [MAPLE PACKAGE] [HTML] [ABSTRACT]
Garvan, Frank; A $q$-product tutorial for a $q$-series MAPLE package, The Andrews Festschrift (Maratea, 1998), Sém. Lothar. Combin. 42 (1999), Art. B42d, 27 pp. ( electronic).
[MR 1701583] [ZBL 1010.11072]

[DVI] [PDF] [ABSTRACT]
Garvan, Frank; Modular functions, Maple and Andrews' 10th problem, Topics in number theory (University Park, PA, 1997), 163--179, Math. Appl., 467, Kluwer Acad. Publ., Dordrecht, 1999.
[MR 1691317] [ZBL 0932.11067]

[ABSTRACT]
Garvan, Frank; Maple V Primer (Release 4), CRC Press, Boca Raton, 1996.
[ZBL 0855.68045]

[DVI] [PDF] [HTML] [ABSTRACT]
J.M. Borwein and F.G. Garvan; Approximations to $\pi$ via the Dedekind eta function. Organic mathematics (Burnaby, BC, 1995), 89--115, CMS Conf. Proc., 20, Amer. Math. Soc., Providence, RI, 1997.
[MR 1483915] [ZBL 893.11050]

[JSTOR] [DVI] [PS] [PDF] [ABSTRACT]
Berndt, Bruce C., Bhargava, S., and Garvan, Frank G.; Ramanujan's theories of elliptic functions to alternative bases, Trans. Amer. Math. Soc. 347 (1995), no. 11, 4163--4244.
Original article: DOI
[MR 1311903] [ZBL 0843.33012]

[DVI] [PDF] [ABSTRACT]
Garvan, Frank G.; A combinatorial proof of the Farkas-Kra theta function identities and their generalizations, J. Math. Anal. Appl. 195 (1995), no. 2, 354--375.
[MR 1354548] [ZBL 0849.11038]

[DVI] [PDF] [ABSTRACT]
Garvan, Frank G.; Ramanujan's theories of elliptic functions to alternative bases---a symbolic excursion, Symbolic computation in combinatorics $\Delta\sb 1$ (Ithaca, NY, 1993), J. Symbolic Comput. 20 (1995), no. 5-6, 517--536.
[MR 1395412] [ZBL 0849.68063]

[JSTOR] [DVI] [PDF] [ABSTRACT]
Borwein, J. M., Borwein, P. B., and Garvan, F. G.; Some cubic modular identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), no. 1, 35--47.
[MR 1243610] [ZBL 0799.33012]

[DVI] [PDF] [ABSTRACT]
Garvan, Frank; Cubic modular identities of Ramanujan, hypergeometric functions and analogues of the arithmetic-geometric mean iteration, The Rademacher legacy to mathematics (University Park, PA, 1992), 245--264, Contemp. Math., 166, Amer. Math. Soc., Providence, RI, 1994.
[MR 1284065] [ZBL 0808.33003]

[DVI] [PS] [PDF] [ABSTRACT]
Garvan, Frank G.; Generalizations of Dyson's rank and non-Rogers-Ramanujan partitions, Manuscripta Math. 84 (1994), no. 3-4, 343--359.
[MR 1291125] [ZBL 0819.11043]

[DVI] [PDF] [ABSTRACT]
Hirschhorn, Michael; Garvan, Frank; and Borwein, Jon; Cubic analogues of the Jacobian theta function $\theta(z,q)$, Canad. J. Math. 45 (1993), no. 4, 673--694.
Original article: https://cms.math.ca/cjm/v45/p673
[MR 1227653] [ZBL 0797.33012]

[DVI] [PDF] [ABSTRACT]
Garvan, Frank G.; Some congruences for partitions that are $p$-cores, Proc. London Math. Soc. (3) 66 (1993), no. 3, 449--478.
Original article: DOI
[MR 1207544] [ZBL 0788.11044]

[PDF] [ABSTRACT]
J. Borwein, P. Borwein and F. Garvan, Hypergeometric analogues of the arithmetic-geometric mean iteration,
Constr. Approx. 9 (1993), no. 4, 509--523.
[MR 1237931] [ZBL 0785.33001]

[DVI] [PDF] [ABSTRACT]
Frank G. Garvan and Gaston H. Gonnet, A proof of the two parameter $q$-cases of the Macdonald-Morris constant term root system conjecture for $S(F\sb 4)$ and $S(F\sb 4)\sp \vee$ via Zeilberger's method, J. Symbolic Comput. 14 (1992), no. 2-3, 141--177.
[MR 1187229] [ZBL 0766.33017]

[PS] [PDF] [ABSTRACT]
Frank G. Garvan and Gaston Gonnet, Macdonald's constant term conjectures for exceptional root systems, Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 2, 343--347.
[MR 1078471] [ZBL 0737.33011]

[DVI] [PDF] [ABSTRACT]
Frank G. Garvan, A number-theoretic crank associated with open bosonic strings, Number theory and cryptography (Sydney, 1989), 221--226, London Math. Soc. Lecture Note Ser., 154, Cambridge Univ. Press, Cambridge, 1990.
[MR 1055413] [ZBL 0717.11043]

[DVI] [PDF] [ABSTRACT]
Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and $t$-cores, Invent. Math. 101 (1990), no. 1, 1--17.
[MR 1055707] [ZBL 0721.11039]

[DVI] [PDF] [ABSTRACT]
F.G. Garvan, A proof of the Macdonald-Morris root system conjecture for $F\sb 4$, SIAM J. Math. Anal. 21 (1990), no. 3, 803--821.
[MR 1046804] [ZBL 0711.33001]

[JSTOR] [DVI] [PDF] [ABSTRACT]
Frank G. Garvan, The crank of partitions mod $8$, $9$ and $10$, Trans. Amer. Math. Soc. 322 (1990), no. 1, 79--94.
[MR 1012520] [ZBL 0717.11042]

[DVI] [PDF] [ABSTRACT]
Frank G. Garvan, Some Macdonald-Mehta integrals by brute force, $q$-series and partitions (Minneapolis, MN, 1988), 77--98, IMA Vol. Math. Appl., 18, Springer, New York, 1989.
[MR 1019845] [ZBL 0697.33008]

[JSTOR] [DVI] [PDF] [ABSTRACT]
Frank Garvan, and Dennis Stanton, Sieved partition functions and $q$-binomial coefficients, Math. Comp. 55 (1990), no. 191, 299--311.
[MR 1023761] [ZBL 0702.05008]

[ABSTRACT]
George E. Andrews and F.G. Garvan, Ramanujan's "lost" notebook. VI. The mock theta conjectures, Adv. in Math. 73 (1989), no. 2, 242--255.
Original article: DOI

[MR 987276] [ZBL 0677.10010]

[DVI] [PDF] [ABSTRACT]
F.G. Garvan, A beta integral associated with the root system $G\sb 2$, SIAM J. Math. Anal. 19 (1988), no. 6, 1462--1474.
[MR 965267] [ZBL 0657.33001]

[HTML] [ABSTRACT]
George E. Andrews and F.G. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 167--171.
Original article: [PDF]
[MR 929094] [ZBL 0646.10008]

[DVI] [PDF] [ABSTRACT]
F.G. Garvan, Combinatorial interpretations of Ramanujan's partition congruences, Ramanujan revisited (Urbana-Champaign, Ill., 1987), 29--45, Academic Press, Boston, MA, 1988.
[MR 938958] [ZBL 0652.10010]

[JSTOR] [ABSTRACT]
F.G. Garvan, New combinatorial interpretations of Ramanujan's partition congruences mod $5,7$ and $11$, Trans. Amer. Math. Soc. 305 (1988), no. 1, 47--77.
[MR 920146] [ZBL 0641.10009]

[HTML]
F.G. Garvan, Generalizations of Dyson's Rank , Ph.D. Thesis, Pennsylvania State University, 1986.

[PDF] [ABSTRACT]
F.G. Garvan, A simple proof of Watson's partition congruences for powers of $7$, J. Austral. Math. Soc. Ser. A 36 (1984), no. 3, 316--334.
Original article: DOI
[MR 733905] [ZBL 0533.10006]

[HTML]
F.G. Garvan, Some Congruence Properties of the Partition Function , M. Sc. Thesis, University of New South Wales, 1982.

Created by F.G. Garvan (fgarvan@math.ufl.edu) on June 17, 1995.
Last update made Sun Apr 7 11:59:18 CDT 2024.

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