Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 72, No. 4, pp. 1183-1189, 2022

Truncations of Gauss' square exponent theorem

Ji-Cai Liu, Shan-Shan Zhao

Received November 17, 2021. Published online June 6, 2022.

Abstract: We establish two truncations of Gauss' square exponent theorem and a finite extension of Euler's identity. For instance, we prove that for any positive integer $n$, $\sum_{k=0}^n(-1)^k \left[ \matrix 2n-k\\ k \right] (q;q^2)_{n-k} q^{{k+1\choose2}} =\sum_{k=-n}^n(-1)^kq^{k^2}$, where $\left[ \matrix n\\ m \right] =\prod_{k=1}^m\frac{1-q^{n-k+1}}{1-q^k}$ and $(a;q)_n=\prod_{k=0}^{n-1}(1-aq^k)$.

Keywords: Gauss' identity; $q$-binomial coefficient; $q$-binomial theorem

Classification MSC: 11B65, 33D15

DOI: 10.21136/CMJ.2022.0429-21

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] G. E. Andrews: The Theory of Partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1998). DOI 10.1017/CBO9780511608650 | MR 1634067 | Zbl 0996.11002
[2] G. E. Andrews, M. Merca: The truncated pentagonal number theorem. J. Comb. Theory, Ser. A 119 (2012), 1639-1643. DOI 10.1016/j.jcta.2012.05.001 | MR 2946378 | Zbl 1246.05014
[3] A. Berkovich, F. G. Garvan: Some observations on Dyson's new symmetries of partitions. J. Comb. Theory, Ser. A 100 (2002), 61-93. DOI 10.1006/jcta.2002.3281 | MR 1932070 | Zbl 1016.05003
[4] S. Chern: Note on the truncated generalizations of Gauss's square exponent theorem. Rocky Mt. J. Math. 48 (2018), 2211-2222. DOI 10.1216/RMJ-2018-48-7-2211 | MR 3892131 | Zbl 1454.11044
[5] W. Chu, L. Di Claudio: Classical Partition Identities and Basic Hypergeometric Series. Quadermi di Matematica 6. Universita degli Studi di Lecce, Lecce (2004). Zbl 1275.11133
[6] C.-Y. Gu, V. J. W. Guo: $q$-analogues of two supercongruences of Z.-W. Sun. Czech. Math. J. 70 (2020), 757-765. DOI 10.21136/CMJ.2020.0516-18 | MR 4151703 | Zbl 07250687
[7] V. J. W. Guo, J. Zeng: Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas. Discrete Math. 308 (2008), 4069-4078. DOI 10.1016/j.disc.2007.07.106 | MR 2427740 | Zbl 1156.05003
[8] V. J. W. Guo, J. Zeng: Two truncated identities of Gauss. J. Comb. Theory, Ser. A 120 (2013), 700-707. DOI 10.1016/j.jcta.2012.12.004 | MR 3007145 | Zbl 1259.05020
[9] M. E. H. Ismail, D. Kim, D. Stanton: Lattice paths and positive trigonometric sums. Constr. Approx. 15 (1999), 69-81. DOI 10.1007/s003659900097 | MR 1660081 | Zbl 0924.42004
[10] J.-C. Liu: Some finite generalizations of Euler's pentagonal number theorem. Czech. Math. J. 67 (2017), 525-531. DOI 10.21136/CMJ.2017.0063-16 | MR 3661057 | Zbl 1458.05025
[11] J.-C. Liu: Some finite generalizations of Gauss's square exponent identity. Rocky Mt. J. Math. 47 (2017), 2723-2730. DOI 10.1216/RMJ-2017-47-8-2723 | MR 3760315 | Zbl 1434.11055
[12] J.-C. Liu, Z.-Y. Huang: A truncated identity of Euler and related $q$-congruences. Bull. Aust. Math. Soc. 102 (2020), 353-359. DOI 10.1017/S0004972720000301 | MR 4176678 | Zbl 1453.05013
[13] R. Mao: Proofs of two conjectures on truncated series. J. Comb. Theory, Ser. A 130 (2015), 15-25. DOI 10.1016/j.jcta.2014.10.004 | MR 3280682 | Zbl 1316.11092
[14] D. Shanks: A short proof of an identity of Euler. Proc. Am. Math. Soc. 2 (1951), 747-749. DOI 10.1090/S0002-9939-1951-0043808-6 | MR 0043808 | Zbl 0044.28403

Affiliations: Ji-Cai Liu (corresponding author), Shan-Shan Zhao, Department of Mathematics, Wenzhou University, No. 586 Meiquan Road, Wenzhou 325035, P. R. China, e-mail: jcliu2016@gmail.com, szhao2021@foxmail.com

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