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Truncated versions of three identities of Euler and Gauss

Part of: Combinatorics Additive number theory; partitions

Published online by Cambridge University Press:  30 August 2022

Olivia X.M. Yao
Olivia X.M. Yao*
Affiliation:
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, Jiangsu Province 215009, P. R. China ([email protected])
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Abstract

In 2012, Andrews and Merca proved a truncated theorem on Euler's pentagonal number theorem. Motivated by the works of Andrews and Merca, Guo and Zeng deduced truncated versions for two other classical theta series identities of Gauss. Very recently, Xia et al. proved new truncated theorems of the three classical theta series identities by taking different truncated series than the ones chosen by Andrews–Merca and Guo–Zeng. In this paper, we provide a unified treatment to establish new truncated versions for the three identities of Euler and Gauss based on a Bailey pair due to Lovejoy. These new truncated identities imply the results proved by Andrews–Merca, Wang–Yee, and Xia–Yee–Zhao.

Keywords

partitionstheta functionstruncated sumspartition-theoretic interpretationsBailey pair

MSC classification

Primary: 11P81: Elementary theory of partitions
Secondary: 05A17: Partitions of integers
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 3 , August 2022 , pp. 775 - 798
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Andrews, G. E., $q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra. C.B.M.S. Regional Conference Series in Math, vol. 66. American Math. Soc., Providence, 1986CrossRefGoogle Scholar
Andrews, G. E., The Theory of Partitions, (Reading, Mass, Addison-Wesley, 1976). reprinted, Cambridge University Press, 1998.Google Scholar
Andrews, G. E. and Merca, M., The truncated pentagonal number theorem, J. Comb. 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. Comb. Theory, Ser. A 154 (2018), 610–619.CrossRefGoogle Scholar
Ballantine, C., Merca, M., Passary, D. and Yee, A. J., Combinatorial proofs of two truncated theta series theorems, J. Combin. Theory, Ser. A 160 (2018), 168–185.CrossRefGoogle Scholar
Burnette, M. and Kolitsch, L., Interpreting the truncated pentagonal number theorem using partition pairs, Electron. J. Comb. 22 (2015), 55.Google Scholar
Chan, S. H., Ho, T. P. N. and Mao, R., Truncated series from the quintuple product identity, J. Num. Theory 169 (2016), 420–438.CrossRefGoogle Scholar
Chern, S., A further look at the truncated pentagonal number theorem, Acta Arith. 189 (2019), 397–403.CrossRefGoogle Scholar
Corteel, S. and Lovejoy, J., Overpartitions, Trans. Amer. Math. Soc. 356 (2004), 1623–1635.CrossRefGoogle Scholar
Gasper, G. and Rahman, M., Basic Hypergeometric Series, Encycl. Math. Appl. (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Guo, V. J. W. and Zeng, J., Two truncated identities of Gauss, J. Comb. Theory, Ser. A 120 (2013), 700–707.CrossRefGoogle Scholar
Hirschhorn, M. D. and Sellers, J. A., Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22 (2010), 273–284.CrossRefGoogle Scholar
Kolitsch, L. W., Generalizations of the truncated pentagonal number theorem results, Ramanujan J. doi: 10.1007/s11139-021-00544-2Google Scholar
Liu, Z.-G., On the $q$-partial differential equations and $q$-series, in: The Legacy of Srinivasa Ramanujan, in: Ramanujan Math. Soc. Lect. Notes Ser., Vol. 20 (Ramanujan Math. Soc., Mysore, 2013), pp. 213–250.Google Scholar
Lovejoy, J., Ramanujan-type partial theta identities and conjugate Bailey pairs, Ramanujan J. 29 (2012), 51–67.CrossRefGoogle Scholar
Mao, R., Proofs of two conjectures on truncated series, J. Comb. Theory, Ser. A 130 (2015), 15–25.CrossRefGoogle Scholar
Mao, R., Some new expansions for certain truncated $q$-series, Ramanujan J. 46 (2018), 475–481.CrossRefGoogle Scholar
Merca, M., Rank partition functions and truncated theta identities, Appl. Anal. Discrete Math. doi:10.2298/AADM190401023MGoogle Scholar
Merca, M., Truncated theta series and Rogers-Ramanujan functions, Exp. Math. 30 (2021), 364–371.CrossRefGoogle Scholar
Merca, M., On two truncated quintuple series theorems, Exp. Math. 31 (2022), 606–610.CrossRefGoogle Scholar
Merca, M. and Yee, A. J., On the sum of parts with multiplicity at least 2 in all the partitions of $n$, Int. J. Number Theory 17 (2021), 665–681.CrossRefGoogle Scholar
Merca, M., Wang, C. and Yee, A. J., A truncated theta identity of Gauss and overpartitions into odd parts, Ann. Comb. 23 (2019), 907–915.CrossRefGoogle Scholar
Seo, S. and Yee, A. J., Enumeration of partitions with prescribed successive rank parity blocks, J. Comb. Theory, Ser. A 158 (2018), 12–35.CrossRefGoogle Scholar
Seo, S. and Yee, A. J., Singular overpartitions and partitions with prescribed hook differences, Ann. Comb. 23 (2019), 1039–1072.CrossRefGoogle Scholar
Wang, C. and Yee, A. J., Truncated Hecke-Rogers type series II, Ramanujan J. doi:10.1007/s11139-021-00407-wGoogle Scholar
Wang, C. and Yee, A. J., Truncated Jacobi triple product series, J. Comb. Theory, Ser. A 166 (2019), 382–392.CrossRefGoogle Scholar
Wang, C. and Yee, A. J., Truncated Hecke-Rogers type series, Adv. Math. 365 (2020), 107051.CrossRefGoogle Scholar
Xia, E. X. W., Infinite families of Hecke-Rogers type series and their truncated representations, Adv. Appl. Math. 137 (2022), 102310.CrossRefGoogle Scholar
Xia, E. X. W., Yee, A. J. and Zhao, X., New truncated theorems for three classical theta function identities, European J. Comb. 101 (2022), 103470.CrossRefGoogle Scholar
Xia, E. X. W. and Zhao, X., Truncated sums for the partition function and a problem of Merca, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 116 (2022), 22.CrossRefGoogle Scholar
Yao, O. X. M., Truncated sums for Hecke-Rogers type identities and Andrews-Hickerson's Bailey pair, Adv. Appl. Math. 140 (2022), 102388.CrossRefGoogle Scholar
Yee, A. J., A truncated Jacobi triple product theorem, J. Comb. Theory, Ser. A 130 (2015), 1–14.CrossRefGoogle Scholar
Zhang, W. J., Further extensions of some truncated Hecke type identities, Acta Math. Sci. 42 (2022), 73–90.CrossRefGoogle Scholar