q-series tools
Definitions and Preliminaries
[OralHistory] Abbate, J. (2001). Oral History: Interview with Lucy Slater. http://ethw.org/Oral-History:Lucy_Slater
[A48] Alder, H. L. (1948). The nonexistence of certain identities in the theory of partitions and compositions. Bull. Amer. Math. Soc., 54(8), 712–722. https://www.ams.org/journals/bull/1948-54-08/S0002-9904-1948-09062-0/S0002-9904-1948-09062-0.pdf
[A98] Alladi, K. (1998). On a partition theorem of Göllnitz and quartic transformations. J. Number Theory, 69, 153–180. https://doi.org/10.1006/jnth.1997.2215
[AA15] Alladi, K., & Andrews, G. E. (2015). The dual of Göllnitz’s (big) partition theorem. Ramanujan J., 36, 171–201. https://doi.org/10.1007/s11139-014-9617-0
[AAG95] Alladi, K., Andrews, G. E., & Gordon, B. (1995). Refinements and generalizations of Capparelli’s conjecture on partitions. J. Algebra, 174, 636–658. https://doi.org/10.1006/jabr.1995.1144
[AG93] Alladi, K., & Gordon, B. (1993). Generalizations of Schur’s partition theorem. Manuscripta Math., 79, 113–126. https://doi.org/10.1007/BF02568332
[AG95] Alladi, K., & Gordon, B. (1995). Schur’s partition theorem, companions, refinements and generalizations. Trans. Amer. Math. Soc., 347(5), 1591–1608. https://doi.org/10.1090/S0002-9947-1995-1297520-X
[A67] Andrews, G. E. (1967). A generalization of the Göllnitz–Gordon partition theorems. Proc. Amer. Math. Soc., 8, 945–952. https://doi.org/10.1090/S0002-9939-1967-0219497-6
[A68] Andrews, G. E. (1968). On q-difference equations for certain well-poised basic hypergeometric series. Quart. J. Math., 19, 433–447. https://doi.org/10.1093/qmath/19.1.433
[A70] Andrews, G. E. (1970). A polynomial identity which implies the Rogers-Ramanujan identities. Scripta Math., 28, 297–305. https://www.scribd.com/doc/6625528/A-Polynomial-Identity-Which-Implies-the-Rogers-Ramanujan-Identities
[A73] Andrews, G. E. (1973). On the q-analog of Kummer’s theorem and applications. Duke Math. J., 40(3), 525–528. https://doi.org/10.1215/S0012-7094-73-04045-3
[A74] Andrews, G. E. (1974a). An analytic generalization of the Rogers–Ramanujan identities for odd moduli. Proc. Nat. Acad. Sci., 71(10), 4082–4085.
[A74m] Andrews, G. E. (1974b). On the general Rogers–Ramanujan theorem. Memoirs Amer. Math. Soc., 152, 86 pp.
[A75] Andrews, G. E. (1975). On Rogers-Ramanujan type identities related to the modulus 11. Proc. London Math. Soc., 30, 330–346. https://doi.org/10.1112/plms/s3-30.3.330
[A76] Andrews, G. E. (1976). The Theory of Partitions (Vol. 2). Addison–Wesley.
[A79] Andrews, G. E. (1979). Partitions: Yesterday and Today. New Zealand Mathematical Society.
[A81] Andrews, G. E. (1981). Ramanujan’s “lost” notebook II: ϑ-function expansions. Adv. Math., 41, 173–185. https://doi.org/10.1016/0001-8708(81)90014-1
[A82] Andrews, G. E. (1982). L. J. Rogers and the Rogers–Ramanujan identities. Math. Chronicle, 11, 1–15.
[A84] Andrews, G. E. (1984). Multiple series Rogers–Ramanujan type identities. Pacific J. Math, 114, 267–283. https://doi.org/10.2140/pjm.1984.114.267
[A85] Andrews, G. E. (1985). Combinatorics and Ramanujan’s “lost” notebook. Surveys in Combinatorics, 103, 1–23.
[A86] Andrews, G. E. (1986a). q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra (Vol. 66). American Mathematical Society. https://bookstore.ams.org/cbms-66
[A86a] Andrews, G. E. (1986b). The fifth and seventh order mock theta functions. Trans. Amer. Math. Soc., 293, 113–134. https://doi.org/10.1090/S0002-9947-1986-0814916-2
[A89] Andrews, G. E. (1989). Physics, Ramanujan, and computer algebra. In D. V. Chudnovsky & R. D. Jenks (Eds.), Computer Algebra. Marcel Dekker, Inc.
[A90] Andrews, G. E. (1990a). Euler’s `exemplum memorabile inductionis fallis’ and q-trinomial coefficients. J. Amer. Math. Soc., 3, 653–669. https://doi.org/10.2307/1990932
[A90qT] Andrews, G. E. (1990b). q-trinomial coefficients and the Rogers–Ramanujan identities. In B. C. B. et al. (Ed.), Analytic Number Theory (pp. 1–11). Birkhauser.
[A94] Andrews, G. E. (1994). Schur’s theorem, Capparelli’s conjecture and q-trinomial coefficients. In G. E. Andrews, D. M. Bressoud, & L. A. Parson (Eds.), The Rademacher Legacy to Mathematics. American Mathematical Society.
[A00] Andrews, G. E. (2001a). Bailey’s transform, lemma, chains and tree. In J. B. et al. (Ed.), Special Functions 2000 (pp. 1–22). Kluwer.
[A01] Andrews, G. E. (2001b). Some debts I owe. In D. Foata & G.-N. Han (Eds.), The Andrews Festschrift: Seventeen Papers on Classical Number Theory and Combinatorics (pp. 1–16). Springer.
[A09] Andrews, G. E. (2009). The finite Heine transformation. Integers, 9 (S), Article 1, 6 pp.
[A13a] Andrews, G. E. (2013a). The discovery of Ramanujan’s lost notebook. In B. C. Berndt & D. Prasad (Eds.), The Legacy of Srinivasa Ramanujan (Vol. 20, pp. 77–88). Ramanujan Mathematical Society, India.
[A13] Andrews, G. E. (2013b). The Selected Works of George E. Andrews (with Commentary). (A. V. Sills, ed.), Imperial College Press, London.
[A14] Andrews, G. E. (2014). Knots and q-series. In K. Alladi, F. G. Garvan, & A. J. Yee (Eds.), Ramanujan 125 (Vol. 627, pp. 17–24). American Mathematical Society.
[AAR99] Andrews, G. E., Askey, R., & Roy, R. (1999). Special Functions (Vol. 71). Cambridge University Press. https://doi.org/10.1017/CBO9781107325937
[AB87] Andrews, G. E., & Baxter, R. J. (1987). Lattice gas generalization of the hard hexagon model III: q-trinomial coefficients. J. Stat. Phys., 47, 297–330. https://doi.org/10.1007/BF01007513
[AB89] Andrews, G. E., & Baxter, R. J. (1989). A Motivated proof of the Rogers–Ramanujan identities. Amer. Math. Monthly, 96, 401–409. https://doi.org/10.1080/00029890.1989.11972207
[AB90] Andrews, G. E., & Baxter, R. J. (1990). SCRATCHPAD explorations for elliptic theta functions. In D. V. Chudnovsky & R. D. Jenks (Eds.), Computers in Mathematics (Vol. 125, pp. 17–33). Marcel Dekker.
[ABF84] Andrews, G. E., Baxter, R. J., & Forrester, P. J. (1984). Eight-vertex SOS model and generalized Rogers–Ramanujan type identities. J. Stat. Phys., 35, 193–266. https://doi.org/10.1007/BF01014383
[AB02] Andrews, G. E., & Berkovich, A. (2002). The WP-Bailey tree and its implications. J. London Math. Soc. (2), 66, 529–549. https://doi.org/10.1112/S0024610702003617
[AB05] Andrews, G. E., & Berndt, B. C. (2005). Ramanujan’s Lost Notebook, Part I. Springer. https://doi.org/10.1007/0-387-28124-X
[AB09] Andrews, G. E., & Berndt, B. C. (2009). Ramanujan’s Lost Notebook, Part II. Springer. https://doi.org/10.1007/b13290
[ABM15] Andrews, G. E., Bringmann, K., & Mahlburg, K. (2015). Double series representations for Schur’s partition function and related identities. J. Combin. Theory Ser. A, 132, 102–119. https://doi.org/10.1016/j.jcta.2014.12.004
[AE04] Andrews, G. E., & Eriksson, K. (2004). Integer partitions. Cambridge University Press. https://doi.org/10.1017/CBO9781139167239
[AKP00] Andrews, G. E., Knofpmacher, A., & Paule, P. (2000). An infinite family of Engel expansions of Rogers–Ramanujan type. Adv. Appl. Math., 25, 2–11. https://doi.org/10.1006/aama.2000.0686
[AS97] Andrews, G. E., & Santos, J. P. O. (1997). Rogers–Ramanujan type identities for partitions with attached odd parts. Ramanujan J., 1, 91–99. https://doi.org/10.1023/A:1009719322309
[Ap48] Apostol, T. M. (1948). A Study of Dedekind Sums and their Generalizations [Phdthesis]. University of California at Berkeley.
[Ap76] Apostol, T. M. (1976). Introduction to Analytic Number Theory. Springer. https://doi.org/10.1007/978-1-4757-5579-4
[Ap97] Apostol, T. M. (1997). Modular functions and Dirichlet series in number theory (2nd ed., Vol. 41). Springer. https://doi.org/10.1007/978-1-4612-0999-7
[AskeyMadisonPerspective] Askey, R. (2001). The work of George Andrews: a Madison perspective. The Andrews Festschrift: Seventeen Papers on Classical Number Theory and Combinatorics, 17–38. https://doi.org/10.1007/978-3-642-56513-7_2
[B35] Bailey, W. N. (1935). Generalized Hypergeometric Series. Cambridge. https://archive.org/details/bwb_P8-ABP-096
[B41] Bailey, W. N. (1941). A note on certain q-identiites. Quart. J. Math., 12, 173–175. https://doi.org/10.1093/qmath/os-12.1.173
[B47] Bailey, W. N. (1947). Some identities in combinatory analysis. Proc. London Math. Soc. (2), 49, 421–435. https://doi.org/10.1112/plms/s2-49.6.421
[B49] Bailey, W. N. (1948). Identities of the Rogers-Ramanujan type. Proc. London Math. Soc. (2), 50, 1–10. https://doi.org/10.1112/plms/s2-50.1.1
[B51] Bailey, W. N. (1951). On the simplification of some identities of the Rogers–Ramanujan type. Proc. London Math. Soc. (3), 1, 217–221. https://doi.org/10.1112/plms/s3-1.1.217
[BF16] Batchelor, M. T., & Foerster, A. (2016). Yang–Baxter integrable models in experiments: from condensed matter to ultracold atoms. J. Physics A, 49(17), 22 pp. https://doi.org/10.1088/1751-8113/49/17/173001
[rjb80] Baxter, R. J. (1980). Hard hexagons: exact solution. J. Phys. A, 13, L61–L70. https://doi.org/10.1088/0305-4470/13/3/007
[rjb81] Baxter, R. J. (1981). Rogers–Ramanujan identities in the hard hexagon model. J. Stat. Phys., 26, 427–452. https://doi.org/10.1007/BF01011427
[B82] Baxter, R. J. (1982). Exactly solved models in statistical mechanics. Academic Press. https://physics.anu.edu.au/research/ftp/_files/Exactly.pdf
[BP83] Baxter, R. J., & Pearce, P. A. (1983). Hard squares with diagonal entries. J. Phys. A, 16, 2239–2255.
[BP84] Baxter, R. J., & Pearce, P. A. (1984). Deviations from the critical density in the generalized hard hexagon model. J. Phys. A, 17, 2095–2108. https://doi.org/10.1088/0305-4470/17/10/021
[BM96] Berkovich, A., & McCoy, B. M. (1996). Continued fractions and fermionic representations for characters of M(p,p’) minimal models. Lett. Math. Phys., 37, 49–66. https://doi.org/10.1007/BF00400138
[BM97] Berkovich, A., & McCoy, B. M. (1997). Generalizations of the Andrews–Bressoud identities for the N=1 superconformal model SM(2,4ν). Math. Comput. Modelling, 26, 37–49. https://doi.org/10.1016/S0895-7177(97)00198-2
[BM98] Berkovich, A., & McCoy, B. M. (1998). Rogers–Ramanujan identities: a century of progress from mathematics to physics. Doc. Math., 163–172. https://doi.org/10.4171/DMS/1-3/17
[BMO96] Berkovich, A., McCoy, B. M., & Orrick, W. P. (1996). Polynomial identities, indices, and duality for the N=1 superconformal model SM(2,4ν). J. Stat. Phys., 83, 795–837. https://doi.org/10.1007/BF02179546
[BMS98] Berkovich, A., McCoy, B. M., & Schilling, A. (1998). Rogers–Schur–Ramanujan type identities for the M(p,p’) minimal models of conformal field theory. Comm. Math. Phys., 191, 211–223. https://doi.org/10.1007/s002200050271
[BR95] Berndt, B. C., & Rankin, R. A. (1995). Ramanujan: Letters and Commentary (Vol. 9). American Mathematical Society. https://bookstore.ams.org/view?ProductCode=HMATH/9
[B03] Bos, M. K. (2003). Coding the principal character formula for affine Kac–Moody Lie algebras. Math. Comput., 72(244), 2001–2012. https://www.jstor.org/stable/4100033
[BP06] Boulet, C., & Pak, I. (2006). A combinatorial proof of the Rogers–Ramanujan and Schur identities. J. Combin. Theory Series A, 113, 119–130. https://doi.org/10.1016/j.jcta.2005.09.007
[BMS09] Bowman, D., McLaughlin, J., & Sills, A. V. (2009). Some more identities of Rogers–Ramanujan type. Ramanujan J., 18, 307–325. https://doi.org/10.1007/s11139-007-9109-6
[B79] Bressoud, D. M. (1979). A generalization of the Rogers–Ramanujan identities for all moduli. J. Combin. Theory, Ser. A, 27, 64–68. https://doi.org/10.1016/0097-3165(79)90008-6
[B80] Bressoud, D. M. (1980a). A combinatorial proof of Schur’s 1926 partition theorem. Proc. Amer. Math. Soc., 1979(2), 338–340. https://doi.org/10.1090/S0002-9939-1980-0565367-X
[B80m] Bressoud, D. M. (1980b). Analytic and combinatorial generalizations of the Rogers–Ramanujan identities. Mem. Amer. Math. Soc., 227, 54 pp. https://www.ams.org/books/memo/0227/
[B81] Bressoud, D. M. (1981). Some identities for terminating q-series. Math. Proc. Cambridge Philos. Soc., 81, 211–223. https://doi.org/10.1017/S0305004100058114
[BZ82] Bressoud, D. M., & Zeilberger, D. (1982). A short Rogers–Ramanujan bijection. Discrete Math., 38, 313–315. https://doi.org/10.1016/0012-365X(82)90298-9
[BO13] Bruinier, J., & Ono, K. (2013). Algebraic formulas for the coefficients of half-integral weight harmonic weak Maas forms. Adv. Math., 246, 198–219. https://doi.org/10.1016/j.aim.2013.05.028
[C96] Capparelli, S. (1996). A construction of the level 3 modules for the affine algebra A₂(2) and a new combinatorial identity of the Rogers–Ramanujan type. Trans. Amer. Math. Soc., 348 (2), 481–501. https://doi.org/10.1090/S0002-9947-96-01535-8
[C59] Carlitz, L. (1959). Some formulas related to the Rogers–Ramanujan identities. Annali Di Matematica Pura et Applicata, 47(1), 243–251. https://doi.org/10.1007/BF02410899
[C05] Carter, R. (2005). Lie algebras of finite and affine type. Cambridge University Press. https://doi.org/10.1017/CBO9780511614910
[C84] Cayley, A. (1884). Note on a partition-series. Amer. J. Math., 6, 63–64. https://doi.org/10.2307/2369210
[CH97] Chan, H. H., & Huang, S.-S. (1997). Ramanujan–Göllnitz–Gordon continued fraction. Ramanujan J., 1, 75–90. https://doi.org/10.1023/A:1009767205471
[CS15] Choliy, Y., & Sills, A. V. (2016). A formula for the partition function that `counts’. Ann. Combin., 20, 301–316. https://doi.org/10.1007/s00026-016-0305-1
[C04] Cigler, J. (2004). q-Fibonacci polynomials and the Rogers–Ramanujan identities. Ann. Combin., 8, 269–285. https://doi.org/10.1007/s00026-004-0220-8
[C06] Cooper, S. (2006). The quintuple product identity. Int. J. Number Theory, 2(1), 115–161. https://doi.org/10.1142/S1793042106000401
[CL04] Corteel, S., & Lovejoy, J. (2004). Overpartitions. Trans. Amer. Math. Soc., 356, 1623–1635. https://doi.org/10.1090/S0002-9947-03-03328-2
[CL09] Corteel, S., & Lovejoy, J. (2009). Overpartitions and the q-Bailey identity. Proc. Edinburgh Math. Soc., 52, 297–306. https://doi.org/10.1017/S0013091507000314
[CSS12] Corteel, S., Savage, C., & Sills, A. V. (2012). Lecture hall sequences, q-series, and asymmetric partition identities. In Partitions, q-series, and Modular Forms (Vol. 23). Springer. https://doi.org/10.1007/978-1-4614-0028-8_6
[7author] Coulson, B., Kanade, S., Lepowsky, J., McRae, R., Qi, F., Russell, M. C., & Sadowski, C. (2017). A motivated proof of the Göllnitz–Gordon–Andrews identities. Ramanujan J., 42(1), 97–129. https://doi.org/10.1007/s11139-015-9722-8
[D42] Daum, J. A. (1942). The basic analogue of Kummer’s theorem. Bull. Amer. Math. Soc., 48, 711–713. https://doi.org/10.1090/s0002-9904-1942-07764-0
[VP96] de la Vallé-Poussin, C. (1896). Recherches analytiques sur la théorie des nombres premiers. Ann. Soc. Sci. Bruxelles, 20, 183–256, 281–297.
[DS05] Diamond, F., & Shurman, J. (2005). A First Course in Modular Forms (Vol. 228). Springer–Verlag. https://doi.org/10.1007/978-0-387-27226-9
[D14] Duke, W. (2014). Almost a century of answering the question: What is a mock theta function? Notices Amer. Math. Soc., 61, 1314–1320.
[D43] Dyson, F. J. (1943). Three identities in combinatory analysis. J. London Math. Soc., 18, 35–39. https://doi.org/10.1112/jlms/s1-18.1.35
[D88] Dyson, F. J. (1988). A walk through Ramanujan’s garden. In G. E. A. et al. (Ed.), Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana–Champaign, June 1–5, 1987 (pp. 7–28). Academic Press.
[ET90] Ekhad, S. B., & Tre, S. (1990). A purely verification proof of the first Rogers–Ramanujan identity. J. Comb. Theory, Ser. A, 54, 309–311. https://doi.org/10.1016/0097-3165(90)90038-X
[E13] Engel, F. (1913). Entwicklung der Zahlen nach Stammbruechen. Verhandlungen Der 52. Versammlung Deutscher Philogen Und Schulmaenner in Marburg, 190–191.
[E48] Euler, L. (1988). Introduction to analysis of the infinite (translation of Introductio in anlysin infinitorum, 1748) (Vol. 1). Springer Verlag. https://doi.org/10.1007/978-1-4612-1021-4
[FOR13] Folsom, A., Ono, K., & Rhoades, R. C. (2013). Mock theta functions and quantum modular forms. Forum of Mathematics, Pi, 1, 27 pp. https://doi.org/10.1017/fmp.2013.3
[FB85] Forrester, P. J., & Baxter, R. J. (1985). Further exact solutions of the eight vertex SOS model and generalized Rogers–Ramanujan type identities. J. Stat. Phys., 38, 435–472. https://doi.org/10.1007/BF01010471
[F16] Fricke, R. (1916). Die Elliptischen Funktionen und ihre Anwendungen. Ersete Teil, Teubner.
[GL15] Garoufalidis, S., & Le, T. T. Q. (2015). Nahm sums, stability and the colored Jones polynomial. Research Math. Sci., 2, 1–55. https://doi.org/10.1186/2197-9847-2-1
[GIS99] Garrett, K. C., Ismail, M. E. H., & Stanton, D. (1999). Variants of the Rogers–Ramanujan identities. Adv. Appl. Math., 23, 274–299. https://doi.org/10.1006/aama.1999.0658
[GM81a] Garsia, A. M., & Milne, S. C. (1981a). A method for constructing bijections for classical partition identities. Proc. Nat. Acad. Sci. USA, 78, 2026–2028. https://doi.org/10.1073/pnas.78.4.2026
[GM81b] Garsia, A. M., & Milne, S. C. (1981b). A Rogers–Ramanujan bijection. J. Combin. Theory Series A, 31, 289–339. https://doi.org/10.1016/0097-3165(81)90062-5
[G13] Gauss, C. F. (1813). Disquistiones generalies circa seriem infinitam. Comm. Soc. Reg. Sci. Gött. Rec., 2.
[GS83] Gessel, I., & Stanton, D. (1983). Applications of q-Lagrange inversion to basic hypergeometric series. Trans. Amer. Math. Soc., 277, 173–201. https://doi.org/10.1090/S0002-9947-1983-0690047-7
[G83] Glaisher, J. W. L. (1883). A theorem in partitions. Messenger of Math., 12, 158–170.
[G60] Göllnitz, H. (1960). Einfache Partitionen [Mathesis]. Göttingen.
[G67] Göllnitz, H. (1967). Partitionen mit Differenzenbedingungen. J. Reine Angew. Math., 225, 154–190.
[G61] Gordon, B. (1961). A combinatorial generalization of the Rogers–Ramanujan identities. Amer. J. Math., 83(2), 393–399. https://doi.org/10.2307/2372962
[G65] Gordon, B. (1965). Some continued fractions of the Rogers–Ramanujan type. Duke Math. J., 31, 741–748. https://doi.org/10.1215/S0012-7094-65-03278-3
[bG83] Gordon, B. (1983). Sieve-equivalence and explicit bijections. J. Combin. Theory Series A, 34, 90–92. https://doi.org/10.1016/0097-3165(83)90043-2
[GOW16] Griffin, M., Ono, K., & Warnaar, S. O. (2016). A framework of Rogers–Ramanujan identities and their arithmetic properties. Duke Math. J., 165, 1475–1527.
[GP10] Gu, N. S. S., & Prodinger, H. (2010). One-parameter generalizations of Rogers–Ramanujan type identiites. Adv. Appl. Math., 45, 149–196. https://doi.org/10.1016/j.aam.2010.01.002
[H96] Hadamard, J. (1896). Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques. Bull. Soc. Math. France, 24, 199–220. https://www.numdam.org/item/10.24033/bsmf.545.pdf
[H63] Hagis, P. (1963). Partitions into odd summands. Amer. J. Math., 85, 213–222. https://doi.org/10.2307/2373210
[NH36] Hall, N. (1936). An algebraic identity. J. London Math. Soc., 11, 276. https://doi.org/10.1112/jlms/s1-11.4.276
[H36] Hardy, G. H. (1937a). Lectures by Godfrey H. Hardy on the Mathematical work of Ramanujan. Edwards Brothers.
[H37] Hardy, G. H. (1937b). The Indian mathematician Ramanujan. Amer. Math. Monthly, 44(3), 137–155. https://doi.org/10.1080/00029890.1937.11987940
[H40] Hardy, G. H. (1940). Ramanujan: Twelve Lectures Suggested by His Life and Work. Cambridge University Press.
[HR18] Hardy, G. H., & Ramanujan, S. (1918). Asymptotic formulæ in combinatory analysis. Proc. London Math. Soc., 17, 75–115. https://doi.org/10.1112/plms/s2-17.1.75
[H78] Heine, E. (1878). Handbuch der Kugelfunctionen, Theorie und Anwendungen (Vol. 1). Reimer.
[H18] Herschel, J. F. W. (1818). On circulating functions, and on the integration of a class of equations of finite differences into which they enter as coefficients. Philosophical Trans. Royal Soc., 108, 144–168. https://doi.org/10.1098/rstl.1818.0011
[H88] Hickerson, D. (1988). A proof of the mock theta conjectures. Invent. Math., 94, 639–660. https://doi.org//10.1007/BF01394279
[H79] Hirschhorn, M. D. (1979). Some partition theorems of Rogers--Ramanujan type. J. Combin. Theory Ser. A, 27, 33–37. https://doi.org//10.1016/0097-3165(79)90005-0
[H42] Hua, L.-K. (1942). On the number of partitions into unequal parts. Trans. Amer. Math. Soc., 51, 194–201. https://doi.org/10.2307/1989985
[I62] Iverson, K. E. (1962). A Programming Language. Wiley.
[J28] Jackson, F. H. (1928). Examples of a generalization of Euler’s transformation for power series. Mess. Math., 57, 169–187.
[JM85] Jimbo, M., & Miwa, T. (1985). A solvable lattice model and related Rogers–Ramanujan type identities. Physica, 15D, 335–353. https://doi.org/10.1016/S0167-2789(85)80003-8
[K90] Kac, V. (1990). Infinite dimensional Lie algebras (3rd ed.). Cambridge University Press. https://doi.org/10.1007/978-1-4757-1382-4
[KLRS15] Kanade, S., Lepowsky, J., Russell, M. C., & Sills, A. V. (2017). Ghost series and a motivated proof of the Andrews–Bressoud identities. J. Combin. Theory Ser. A, 146, 33–62. https://doi.org/10.1016/j.jcta.2016.07.004
[KR15] Kanade, S., & Russell, M. C. (2015). IdentityFinder and some new identities of Rogers–Ramanujan type. Experimental Mathematics, 24(2), 419–423. https://doi.org/10.1080/10586458.2015.1015186
[K08] Kilford, L. J. P. (2008). Modular Forms: A Classical and Computational Introduction. Imperial College Press.
[KK88] Knopfmacher, A., & Knopfmacher, J. (1988). Inverse polynomial expansions of Laurent series. In Constr. Approx. (Vol. 4, pp. 379–389). https://doi.org/10.1007/BF02075468
[K93] Knopp, M. I. (1993). Modular Functions in Analytic Number Theory (second). AMS Chelsea.
[K92] Knuth, D. E. (1992). Two notes on notation. Amer. Math. Monthly, 99(5), 403–422.
[K84] Koblitz, N. (1984). Introduction to Elliptic Curves and Modular Forms (Vol. 97). Springer–Verlag.
[L40] Lebesgue, V.-A. (1840). Sommation de quelques séries. Journal Math. Pures Appl., 5, 42–71. http://www.numdam.org/item/JMPA_1840_1_5__42_0.pdf
[L46] Lehmer, D. H. (1946). Two nonexistence theorems on partitions. Bull. Amer. Math. Soc., 52, 538–544.
[L41] Lehner, J. (1941). A partition function connected with the modulus 5. Duke Math. J., 8, 631–655. https://doi.org/10.1215/S0012-7094-41-00854-2
[L07] Lepowsky, J. (2007). Some developments in vertex operator algebra theory, old and new. In Y.-Z. Huang & K. C. Misra (Eds.), Lie Algebras, Vertex Operator Algebras and their Applications (pp. 355–388). American Mathematical Society.
[LM78] Lepowsky, J., & Milne, S. (1978). Lie algebraic approaches to classical partition identities. Adv. Math., 29, 15–59. https://doi.org/10.1016/0001-8708(78)90004-X
[LW78] Lepowsky, J., & Wilson, R. L. (1978). Construction of the affine Lie algebra A₁(1). Comm. Math. Phys., 62, 45–53. https://doi.org/10.1007/BF01940329
[LW81] Lepowsky, J., & Wilson, R. L. (1981). A new family of algebras underlying the Rogers–Ramanujan identities. Proc. Nat. Acad. Sci. USA, 78, 7254–7258. https://doi.org/10.1073/pnas.78.12.7254
[LW82] Lepowsky, J., & Wilson, R. L. (1982). A Lie theoretic interpretation and proof of the Rogers–Ramanujan identities. Adv. Math., 45, 21–72. https://doi.org/10.1016/S0001-8708(82)80012-1
[LW84] Lepowsky, J., & Wilson, R. L. (1984). The structure of standard modules I: universal algebras and the Rogers–Ramanujan identities. Invent. Math., 77, 199–290. https://doi.org/10.1007/BF01388447
[LW85] Lepowsky, J., & Wilson, R. L. (1985). The structure of standard modules II: the case A₁(1), principal gradation. Invent. Math., 79, 417–442. https://doi.org/10.1007/BF01388515
[LZ12] Lepowsky, J., & Zhu, M. (2012). A motivated proof of Gordon’s identities. Ramanujan J., 29, 199–211. https://doi.org/10.1007/s11139-012-9388-4
[L84] Loxton, J. H. (1984). Special values of the dilogarithm function. Acta Arith., 43, 155–166. https://doi.org/10.4064/aa-43-2-155-166
[nova] Lynch, J. (original airdate: October 28, 1997). The Proof. Nova, PBS television series.
[M99] Macdonald, I. G. (1999). Symmetric Functions and Hall Polynomials (2nd ed.). Oxford University Press. https://doi.org//10.1093/oso/9780198534891.001.0001
[M18] MacMahon, P. A. (1918). Combinatory Analysis: Vol. II. Cambridge University Press. https://archive.org/details/combinatoryanaly02macmuoft/page/n5/mode/2up
[MS08] McLaughlin, J., & Sills, A. V. (2008). Ramanujan–Slater type identities related to the moduli 18 and 24. J. Math. Anal. Appl., 344, 765–777. https://doi.org/10.1016/j.jmaa.2008.03.033
[MS09] McLaughlin, J., & Sills, A. V. (2009). Combinatorics of Ramanujan–Slater type identities. Integers, 9 (S), Article 10, 14 pp.
[MS12] McLaughlin, J., & Sills, A. V. (2012). On a pair of identities from Ramanujan’s lost notebook. Ann. Comb., 16, 591–607. https://doi.org/10.1007/s00026-012-0148-3
[MSZ08] McLaughlin, J., Sills, A. V., & Zimmer, P. (2008). Rogers–Ramanujan–Slater type identities. Electronic J. Comb., 15, Dynamic Survey #15, 59 pp.
[MSZ09] McLaughlin, J., Sills, A. V., & Zimmer, P. (2009). Rogers–Ramanujan computer searches. J. Symbolic Computation, 44, 1068–1078. https://doi.org/10.1016/j.jsc.2009.02.003
[M54] Meinardus, G. (1954). Asymptotische Aussagen über Partitionen. Math. Z., 59, 388–398. https://doi.org/10.1007/BF01180268
[M94] Melzer, E. (1994). Fermionic character sums and the corner transfer matrix. Internat. J. Modern Phys. A, 9, 1115–1136. https://doi.org/10.1142/S0217751X94000510
[MZ05] Mohammed, M., & Zeilberger, D. (2005). Sharp upper bounds for the orders of the recurrences output by the Zeilberger and q-Zeilberger algorithms. J. Symbolic Comput., 39, 201–207. https://doi.org/10.1016/j.jsc.2004.10.002
[N14] Nandi, D. (2014). Partition identities arising from the standard A₂(2) modules of level 4 [Phdthesis]. Rutgers.
[N40] Niven, I. (1940). On a certain partition function. Amer. J. Math., 353–364. https://doi.org/10.2307/2371459
[O88] O’Hara, K. M. (1988). Bijections for partition identities. J. Combin. Theory Series A, 49, 13–25. https://doi.org/10.1016/0097-3165(88)90026-X
[O03] Ono, K. (2003). The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series (Vol. 103). American Mathematical Society.
[P06] Pak, I. (2006). Partition bijections: a survey. Ramanujan J., 12, 5–75. https://doi.org/10.1007/s11139-006-9576-1
[P82] Paule, P. (1982). Zwei neue Transformationen als elementare Anwendungen der q-Vandermonde Formel [Phdthesis]. University of Vienna.
[P85] Paule, P. (1985). On identities of the Rogers–Ramanujan type. J. Math. Anal. Appl., 107, 255–284. https://doi.org/10.1016/0022-247X(85)90368-3
[P87] Paule, P. (1987). A note on Bailey’s lemma. J. Combin. Theory, Ser. A, 44, 164–167. https://doi.org/10.1016/0097-3165(87)90070-7
[P94] Paule, P. (1994). Short and easy computer proofs of the Rogers–Ramanujan identities and identities of similar type. Electronic J. Combin., 1, 9 pp.
[PWZ96] Petkovšek, M., Wilf, H. S., & Zeilberger, D. (1996). A=B. A. K. Peters, Ltd.
[R37] Rademacher, H. A. (1937). On the partition function p(n). Proc. London Math. Soc. (2), 43, 241–254. https://doi.org/10.1112/plms/s2-43.4.241
[R43] Rademacher, H. A. (1943). On the expansion of the partition function in a series. Ann. Math., 44(3), 416–422. https://doi.org/10.2307/1968973
[R73] Rademacher, H. A. (1973). Topics in Analytic Number Theory. Springer. https://doi.org/10.1007/978-3-642-80615-5
[R27] Ramanujan, S. (1927). Collected Papers. (G. H. Hardy, P. V. Seshu Aiyar,.
[R88] Ramanujan, S. (1988). The Lost Notebook and Other Unpublished Papers. Narosa Publishing House.
[R82] Remmel, J. (1982). Bijective proofs of some classical partition identities. J. Combin. Theory Series A, 33, 273–286.
[R13] Rhoades, R. C. (2013). On Ramanujan’s defintion of mock theta function. Proc. Nat. Acad. Sci. USA, 110(19), 7592–7594.
[R93] Rogers, L. J. (1893a). Memoir on the expansion of some infinite products. Proc. London Math. Soc. (1), 24, 337–352. https://doi.org/10.1112/plms/s1-24.1.337
[R93a] Rogers, L. J. (1893b). On a three-fold symmetry in the elements of Heine’s series. Proc. London Math. Soc., 24, 171–179.
[R94] Rogers, L. J. (1894). Second memoir on the expansion of certain infinite products. Proc. London Math. Soc. (1), 25, 318–343. https://doi.org/10.1112/plms/s1-25.1.318
[R95] Rogers, L. J. (1895). Third memoir on the expansion of certain infinite products. Proc. London Math. Soc. (1), 26, 15–32. https://doi.org/10.1112/plms/s1-26.1.15
[R17] Rogers, L. J. (1917). On two theorems of combinatory analysis and some allied identities. Proc. London Math. Soc., 16, 315–336.
[RR19] Rogers, L. J., & Ramanujan, S. (1919). Proof of certain identities in combinatory analysis. C Ambr. Phil. Soc. Proc., 19, 211–216. https://doi.org/10.1112/plms/s2-16.1.315
[R09] Rota, G.-C. (2009). Indiscrete Thoughts. Birkhäuser.
[R11] Rothe, H. (1811). Systematisches Lehrbuch der Aritmetik.
[SS11] Savage, C.D. and Sills, A. V. (2011). On an identity of Gessel and Stanton and the new little Göllnitz identities. Adv. Appl. Math, 46, 563–575. https://doi.org/10.1016/j.aam.2009.12.009
[SW98] Schilling, A., & Warnaar, S. O. (1998). Supernomial coefficients, polynomial identities, and q-series. Ramanujan J., 2, 459–494.
[S17] Schur, I. (1917). Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche. S.–B. Preuss. Akad. Wiss. Phys. Math. Klasse, 302–321.
[S26] Schur, I. (1926). Zur additive Zahlentheorie. S.–B. Preuss. Akad. Wiss. Phys. Math. Klasse, 488–495.
[Sears51] Sears, D. B. (1951). On the transformation theory of basic hypergeometric functions. Proc. London Math. Soc. (2), 53, 158–180.
[Sears52] Sears, D. B. (1952). Two identities of Bailey. J. London Math. Soc., 27, 510–511.
[S36] Selberg, A. (1936). Über einige arithmetische Identitäten. Avrandlinger Norske Akad., 8, 23 pp.
[S03] Sills, A. V. (2003). Finite Rogers–Ramanujan type identities. Electronic J. Combin., #R13, 122 pp.
[S04] Sills, A. V. (2004a). q-difference equations and identities of the Rogers–Ramanujan-Bailey type. J. Difference Equ. Appl., 10, 1069–1084.
[S04RRtools] Sills, A. V. (2004b). RRtools—a Maple package for aiding the discovery and proof of finite Rogers–Ramanujan type identities. J. Symbolic Comput., 37, 415–448.
[S05] Sills, A. V. (2005). Identities of the Rogers–Ramanujan–Bailey type. J. Math. Anal. App., 308, 669–688.
[S06] Sills, A. V. (2006). On identities of the Rogers–Ramanujan type. Ramanujan J., 11, 403–429.
[S07] Sills, A. V. (2007). Identities of the Rogers–Ramanujan–Slater type. Int. J. Number Theory, 3, 293–323.
[S51] Slater, L. J. (1951a). A new proof of Rogers’s transformations of infinite series. Proc. London Math. Soc. (2), 53, 460–475.
[S51a] Slater, L. J. (1951b). Functions of hypergeometric type [Phdthesis]. University of London, Bedford College.
[S52] Slater, L. J. (1952). Further identities of the Rogers-Ramanujan type. Proc. London Math. Soc. (2), 54, 147–167. https://doi.org/10.1112/plms/s2-54.2.147
[S54] Slater, L. J. (1954). A note on equivalent product theorems. Math. Gazette, 38, 127–128.
[S60] Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge University Press.
[S62] Slater, L. J. (1962). Wilfrid Norman Bailey. J. London Math. Soc., 37, 504–512.
[S66] Slater, L. J. (1966). Generalized Hypergeometric Functions. Cambridge University Press.
[S16] Smoot, N. A. (2016). A partition function connected with the Göllnitz–Gordon identities. In M. C. Pereyra, S. Marcantongnini, A. M. Stokolos, & W. U. Romero (Eds.), Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory: Celebrating Cora Sadosky’s Life. Springer.
[S01] Stanton, D. (2001). The Bailey–Rogers–Ramanujan group. Q-Series with Applications in Combinatorics, Number Theory, and Physics, 291.
[S31] Starcher, G. W. (1931). On identities arising from solutions to q-difference equations and some interpretations in number theory. Amer. J. Math., 53, 801–816.
[Stein07] Stein, W. A. (2007). Modular Forms, a Computational Approach (Vol. 79). American Mathematical Society.
[S90] Stembridge, J. R. (1990). Hall–Littlewood functions, Plane partitions, and the Rogers–Ramanujan identities. Trans. Amer. Math. Soc., 319(2), 469–498.
[S82] Sylvester, J. J., & Franklin, F. (1882). A constructive theory of partitions, arranged in three acts, an interact, and an exodion. Amer. J. Math., 5(1), 251–330.
[TC95] Tamba, M., & Xie, C. (1995). Level three standard modules for A₂(2). J. Pure Appl. Algebra, 105 (1), 53–92.
[TM96] Tannery, J., & Molk, J. (1896). Eléments de la théorie des Fonctions Elliptiques. Gauthiers–Villars.
[T11] Todt, H. (2011). Asymptotics in partition functions [Phdthesis]. Pennsylvania State University.
[U20] Uspensky, J. V. (1920). Asymptotic formulæ for numerical functions which occur in the theory of partitions. Izv. Akad. Nauk SSSR, 14, 199–218.
[VJ80] Verma, A., & Jain, V. K. (1980). Transformations between basic hypergeometric series on different bases and identities of the Rogers–Ramanujan type. J. Math. Anal. Appl., 76, 230–269.
[VJ82] Verma, A., & Jain, V. K. (1982). Transformations of non-terminating basic hypergeometric series, their contour integrals and applications to Rogers–Ramanujan type identities. J. Math. Anal. Appl., 87, 9–44.
[W55] Wallis, J. (1655). Arithmetica Infinitorum.
[W01b] Warnaar, S. O. (2001a). Refined q-trinomial coefficients and character identities. J. Stat. Phys., 102, 1065–1081.
[W01] Warnaar, S. O. (2001b). The generalized Borwein conjecture. Q-Series with Applications to Combinatorics, Number Theory, and Physics, 291, 243–267.
[W03] Warnaar, S. O. (2003). Partial theta functions I: Beyond the lost notebook. Proc. London Math. Soc., 87, 363–395.
[W29] Watson, G. N. (1929). A new proof of the Rogers–Ramanujan identities. J. London Math. Soc., 4, 4–9.
[W36] Watson, G. N. (1936). The final problem: an account of the mock theta functions. J. London Math. Soc. (1), 11, 55–80.
[W37] Watson, G. N. (1937). The mock theta functions (2). Proc. London Math Soc. (2), 42, 274–304.
[W44] Watson, G. N. (1944). A treatise on the theory of Bessel functions (2nd ed.). Cambridge University Press.
[W26] Whipple, F. J. W. (1926). On well-poised series, generalized hypergeometric series having parameters in pairs, each pair with the same sum. J. London Math. Soc., 24, 247–263.
[WW27] Whittaker, E. T., & Watson, G. N. (1927). A Course of Modern Analysis (4th ed.). Cambridge.
[WZ90] Wilf, H. S., & Zeilberger, D. (1990). Rational functions certify combinatorial identities. J. Amer. Math. Soc., 3, 147–158.
[WZ92] Wilf, H. S., & Zeilberger, D. (1992). An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent. Math., 108, 575–633.
[Z10] Zagier, D. (2010). Quantum modular forms. Clay Mathematics Proceedings, 11, 659–675.
[Z90] Zeilberger, D. (1990). A fast algorithm for proving terminating hypergeometric identities. Discrete Math., 80, 207–211.
[Z91] Zeilberger, D. (1991). The method of creative telescoping. J. Symbolic Comput., 11, 195–204.
[Z39] Zuckerman, H. S. (1939). On the coefficients of certain modular forms belonging to subgroups of the modular group. Trans. Amer. Math. Soc., 45, 298–321.
[Z02] Zwegers, S. (2002). Mock theta functions [Phdthesis]. Univ. Utrecht, the Netherlands.
[Z09] Zwegers, S. (2009). On two fifth order mock theta functions. Ramanujan J., 20, 207–214.