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


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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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