Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T06:59:29.511Z Has data issue: false hasContentIssue false
  • English
  • Français

SUMS OF PARTIAL THETA FUNCTIONS THROUGH AN EXTENDED BAILEY TRANSFORM

Part of: Basic hypergeometric functions Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  13 May 2020

MOHAMED EL BACHRAOUI Open the ORCID record for MOHAMED EL BACHRAOUI [Opens in a new window]
MOHAMED EL BACHRAOUI*
Affiliation:
Department of Mathematical Sciences, United Arab Emirates University, PO Box 15551, Al-Ain, UAE email melbachraoui@uaeu.ac.ae
Get access Rights & Permissions [Opens in a new window]

Abstract

In this note, we evaluate sums of partial theta functions. Our main tool is an application of an extended version of the Bailey transform to an identity of Gasper and Rahman on $q$-hypergeometric series.

Keywords

partial theta functionsBailey transformq-seriesbasic hypergeometric series

MSC classification

Primary: 05A30: $q$-calculus and related topics
Secondary: 11P84: Partition identities; identities of Rogers-Ramanujan type 33D15: Basic hypergeometric functions in one variable, $_rphi_s$
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 1 - 10
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrews, G. E., ‘Ramanujan’s “lost” notebook. I. Partial 𝜃-functions’, Adv. Math. 41 (1981), 137–172.CrossRefGoogle Scholar
Andrews, G. E., ‘Multiple series Rogers–Ramanujan type identities’, Pacific J. Math. 114 (1984), 267–283.10.2140/pjm.1984.114.267CrossRefGoogle Scholar
Andrews, G. E., The Theory of Partitions (Cambridge University Press, Cambridge, 1998).Google Scholar
Andrews, G. E. and Warnaar, S. O., ‘The product of partial theta functions’, Adv. Appl. Math. 39 (2007), 116–120.10.1016/j.aam.2005.12.003CrossRefGoogle Scholar
Bailey, W. N., ‘Some identities in combinatory analysis’, Proc. Lond. Math. Soc. (2) 49 (1947), 421–435.Google Scholar
Bailey, W. N., ‘Identities of the Rogers–Ramanujan type’, Proc. Lond. Math. Soc. (2) 50 (1949), 1–10.Google Scholar
Bressoud, D. M., ‘Some identities for terminating q-series’, Math. Proc. Cambridge Philos. Soc. 81 (1981), 211–223.10.1017/S0305004100058114CrossRefGoogle Scholar
Gasper, G. and Rahman, M., Basic Hypergeometric Series (Cambridge University Press, Cambridge, 2004).10.1017/CBO9780511526251CrossRefGoogle Scholar
Guo, V. J. W. and Krattenthaler, C., ‘Some divisibility properties of binomial and q-binomial coefficients’, J. Number Theory 135 (2014), 167–184.10.1016/j.jnt.2013.08.012CrossRefGoogle Scholar
Kim, B. and Lovejoy, J., ‘Ramanujan-type partial theta identities and conjugate Bailey pairs, II: multisums’, Ramanujan J. 46 (2018), 743–764.10.1007/s11139-017-9919-0CrossRefGoogle Scholar
Lovejoy, J., ‘Ramanujan-type partial theta identities and conjugate Bailey pairs’, Ramanujan J. 29 (2012), 51–67.10.1007/s11139-011-9356-4CrossRefGoogle Scholar
Ramanujan, S., The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988).Google Scholar
Slater, L. J., ‘A new proof of Rogers’ transformations of infinite series’, Proc. Lond. Math. Soc. (2) 53 (1951), 460–475.CrossRefGoogle Scholar
Slater, L. J., ‘Further identities of the Rogers–Ramanujan type’, Proc. Lond. Math. Soc. (2) 54 (1952), 147–167.CrossRefGoogle Scholar
Slater, L. J., Generalized Hypergeometric Functions (Cambridge University Press, Cambridge, 1966).Google Scholar
Sun, L. H., ‘An extension of the Andrews–Warnaar partial theta function identity’, Adv. Appl. Math. 115 (2020), Article ID 101985.10.1016/j.aam.2019.101985CrossRefGoogle Scholar
Wang, J. and Ma, X., ‘On the Andrews–Warnaar identities for partial theta functions’, Adv. Appl. Math. 97 (2018), 36–53.10.1016/j.aam.2018.02.002CrossRefGoogle Scholar
Warnaar, S. O., ‘Partial theta functions. I. Beyond the lost notebook’, Proc. Lond. Math. Soc. 87 (2003), 363–395.10.1112/S002461150201403XCrossRefGoogle Scholar
Warnaar, S. O., Partial Theta Functions (Springer, New York, 2019), in press.Google Scholar