Article
| Title: | Truncations of Gauss' square exponent theorem (English) |
| Author: | Liu, Ji-Cai |
| Author: | Zhao, Shan-Shan |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 4 |
| Year: | 2022 |
| Pages: | 1183-1189 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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 $n$, $$ \sum _{k=0}^n(-1)^k \left [ \begin{matrix} 2n-k\\ k \end{matrix} \right ] (q;q^2)_{n-k}q^{{k+1\choose 2}} =\sum _{k=-n}^n(-1)^kq^{k^2}, $$ where $$ \left [ \begin{matrix} n\\ m\end{matrix} \right ] =\prod _{k=1}^m\frac {1-q^{n-k+1}}{1-q^k} \quad \text {and} \quad (a;q)_n=\prod _{k=0}^{n-1}(1-aq^k). $$ (English) |
| Keyword: | Gauss' identity |
| Keyword: | $q$-binomial coefficient |
| Keyword: | $q$-binomial theorem |
| MSC: | 11B65 |
| MSC: | 33D15 |
| idZBL: | Zbl 07655793 |
| idMR: | MR4517606 |
| DOI: | 10.21136/CMJ.2022.0429-21 |
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| Date available: | 2022-11-28T11:43:05Z |
| Last updated: | 2023-04-11 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151140 |
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