Hostname: page-component-5cf477f64f-xc2pj Total loading time: 0 Render date: 2025-03-28T01:32:23.462Z Has data issue: false hasContentIssue false
  • English
  • Français

TWO NEW TRUNCATED IDENTITIES AND PARTITION THEOREMS

Part of: Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  21 March 2025

RENRONG MAO Open the ORCID record for RENRONG MAO [Opens in a new window]
RENRONG MAO*
Affiliation:
Department of Mathematics, Soochow University, SuZhou 215006, PR China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish two new truncated identities for theta series. Their partition-theoretic interpretations are also given.

Keywords

partitionspartitions with distinct odd partsEuler’s pentagonal number theoremGauss’ identities

MSC classification

Primary: 11P84: Partition identities; identities of Rogers-Ramanujan type
Secondary: 05A17: Partitions of integers 05A19: Combinatorial identities, bijective combinatorics
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 8
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of 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.)

Footnotes

This work is supported by NSFC (National Natural Science Foundation of China) through Grant NSFC 12471315.

References

Andrews, G. E., The Theory of Partitions, The Encyclopedia of Mathematics and Its Applications, 2 (Addison-Wesley, New York, 1976).Google Scholar
Andrews, G. E. and Merca, M., ‘The truncated pentagonal number theorem’, J. Combin. Theory Ser. A 119 (2012) 1639–1643.CrossRefGoogle Scholar
Andrews, G. E. and Merca, M., ‘Truncated theta series and a problem of Guo and Zeng’, J. Combin. Theory Ser. A 154 (2018) 610–619.CrossRefGoogle Scholar
Chern, S. and Xia, E. X. W., ‘Two conjectures of Andrews, Merca and Yee on truncated theta series’, J. Combin. Theory Ser. A 206 (2024), Article no. 105874, 16 pages.Google Scholar
Guo, V. J. W. and Zeng, J., ‘Two truncated identities of Gauss’, J. Combin. Theory Ser. A 120 (2013), 700–707.CrossRefGoogle Scholar
He, T. Y., Ji, K. Q. and Zang, W. J. T., ‘Bilateral truncated Jacobi’s identity’, European J. Combin. 51 (2016), 255–267.Google Scholar
Li, X., ‘A generalized truncated sums of Jacobi triple product series and some related truncated theorems’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 117(2) (2023), Article no. 86, 14 pages.Google Scholar
Mao, R., ‘Proofs of two conjectures on truncated series’, J. Combin. Theory Ser. A 130 (2015), 15–25.Google Scholar
Mao, R., ‘Some new expansions for certain truncated $q$ -series’, Ramanujan J. 46(2) (2018), 475–481.CrossRefGoogle Scholar
Merca, M., ‘From sums of divisors to partition congruences’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 118(3) (2024), Article no. 115, 20 pages.Google Scholar
Rogers, L. J., ‘On two theorems of combinatory analysis and some allied identities’, Proc. Lond. Math. Soc. (2) 16 (1916), 315–336.Google Scholar
Wang, C. and Yee, A. J., ‘Truncated Jacobi triple product series’, J. Combin. Theory Ser. A 166 (2019), 382–392.CrossRefGoogle Scholar
Wang, C. and Yee, A. J., ‘Truncated Hecke–Rogers type series’, Adv. Math. 365 (2020), Article no. 107051, 19 pages.Google Scholar
Xia, E. X. W., Yee, A. J. and Zhao, X., ‘New truncated theorems for three classical theta function identities’, European J. Combin. 101 (2022), Article no. 103470, 23 pages.CrossRefGoogle Scholar
Yao, O. X. M., ‘Truncated sums for certain restricted partition functions and transformation formulas for basic hypergeometric series’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 117(3) (2023), Article no. 106, 18 pages.Google Scholar
Yao, O. X. M., ‘Proof of a conjecture of Ballantine and Merca on truncated sums of $6$ -regular partitions’, J. Combin. Theory Ser. A 206 (2024), Article no. 105903, 10 pages.CrossRefGoogle Scholar
Yee, A. J., ‘The truncated Jacobi triple product theorem’, J. Combin. Theory Ser. A 130 (2015), 1–14.Google Scholar