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THE ROGERS–RAMANUJAN CONTINUED FRACTION AND RELATED ETA-QUOTIENT REPRESENTATIONS

Part of: Elementary number theory Additive number theory; partitions

Published online by Cambridge University Press:  17 September 2020

SHANE CHERN Open the ORCID record for SHANE CHERN [Opens in a new window]  and
DAZHAO TANG
SHANE CHERN
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USAshanechern@psu.edu
DAZHAO TANG*
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin300072, P. R.China
*
e-mail: dazhaotang@sina.com
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Abstract

We construct eta-quotient representations of two families of q-series involving the Rogers–Ramanujan continued fraction by establishing related recurrence relations. We also display how these eta-quotient representations can be utilised to dissect certain q-series identities.

Keywords

Rogers–Ramanujan continued fractioneta-quotient representationsrecurrence relations

MSC classification

Primary: 11P84: Partition identities; identities of Rogers-Ramanujan type
Secondary: 11A55: Continued fractions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 248 - 259
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The second author was supported by the Postdoctoral Science Foundation of China (No. 2019M661005) and the Fundamental Research Funds for the Central Universities (No. 2018CDXYST0024).

References

Andrews, G. E. and Berndt, B. C., Ramanujan’s Lost Notebook. Part I (Springer, New York, 2005).10.1007/0-387-28124-XCrossRefGoogle Scholar
Andrews, G. E., Berndt, B. C., Jacobsen, L. and Lamphere, R. L., ‘The continued fractions found in the unorganized portions of Ramanujan’s notebooks’, Mem. Amer. Math. Soc. 99(477), 71.Google Scholar
Apostol, T. M., Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, 41 (Springer-Verlag, New York, 1976).10.1007/978-1-4684-9910-0CrossRefGoogle Scholar
Baruah, N. D. and Begum, N. M., ‘Exact generating functions for the number of partitions into distinct parts’, Int. J. Number Theory 14(7) (2018), 1995–2011.10.1142/S1793042118501191CrossRefGoogle Scholar
Baruah, N. D., Begum, N. M. and Das, H., ‘Results on some partition functions arising from certain relations involving the Rogers–Ramanujan continued fractions’, Preprint, arXiv:2005.06799.Google Scholar
Berndt, B. C., Number Theory in the Spirit of Ramanujan (American Mathematical Society, Providence, RI, 2004).Google Scholar
Chern, S., ‘1-Shell totally symmetric plane partitions (TSPPs) modulo powers of $5$ ’, Ramanujan J., to appear.Google Scholar
Chern, S. and Hirschhorn, M. D., ‘Partitions into distinct parts modulo powers of $5$ ’, Ann. Comb. 23(3–4) (2019), 659–682.10.1007/s00026-019-00439-6CrossRefGoogle Scholar
Chern, S. and Tang, D., ‘Elementary proof of congruences modulo 25 for broken $k$ -diamond partitions’, Preprint, arXiv:1807.01890.Google Scholar
Cooper, S., Ramanujan’s Theta Functions (Springer, Cham, 2017).CrossRefGoogle Scholar
Gugg, C., ‘Modular equations for cubes of the Rogers–Ramanujan and Ramanujan– Göllnitz–Gordon functions and their associated continued fractions’, J. Number Theory 132(7) (2012), 1519–1553.10.1016/j.jnt.2012.01.005CrossRefGoogle Scholar
Hirschhorn, M. D., The Power of $q$ . A Personal Journey, Developments in Mathematics, 49 (Springer, Cham, 2017).CrossRefGoogle Scholar
Newman, M., ‘Construction and application of a class of modular functions. II’, Proc. London Math. Soc. 9(3) (1959), 373–387.10.1112/plms/s3-9.3.373CrossRefGoogle Scholar
Ramanujan, S., Notebooks , Vol. II (Tata Institute of Fundamental Research, Bombay 1957; reprinted by Springer-Verlag, Berlin, 1984).Google Scholar
Ramanujan, S., The Lost Notebook and Other Unpublished Papers (Narosa Publishing House, New Delhi, 1988).Google Scholar
Rogers, L. J., ‘Second memoir on the expansion of certain infinite products’, Proc. Lond. Math. Soc. 25 (1894), 318–343.Google Scholar
Rogers, L. J., ‘On a type of modular relation’, Proc. London Math. Soc. (2) 19(5) (1921), 387–397.10.1112/plms/s2-19.1.387CrossRefGoogle Scholar
Schur, I., ‘Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche’, Berliner Sitzungsberichte 23 (1917), 301–321.Google Scholar
Tang, D., ‘New congruences for broken $k$ -diamond partitions’, J. Integer Seq. 21(2) (2018), Article ID 18.5.8.Google Scholar
Tang, D., ‘Congruences for overpartition pairs and 5 dots bracelet partitions modulo $25$ ’, Integers 20 (2020), Article ID A28.Google Scholar
Watson, G. N., ‘Theorems stated by Ramanujan (VII): theorems on continued fractions’, J. London Math. Soc. 4(1) (1929), 39–48.CrossRefGoogle Scholar
Zhang, W. and Shi, J., ‘Congruences for the coefficients of the mock theta function $\beta (q)$ ’, Ramanujan J. 49(2) (2019), 257–267.CrossRefGoogle Scholar