Truncations of Gauss' square exponent theorem

Liu, Ji-Cai, and Zhao, Shan-Shan. "Truncations of Gauss' square exponent theorem." Czechoslovak Mathematical Journal 72.4 (2022): 1183-1189. <http://eudml.org/doc/298941>.

@article{Liu2022,
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 $n$, \[ \sum \_\{k=0\}^n(-1)^k \left[ \begin\{matrix\} 2n-k\\ k \end\{matrix\} \right] (q;q^2)\_\{n-k\}q^\{\{k+1\atopwithdelims ()2\}\} =\sum \_\{k=-n\}^n(-1)^kq^\{k^2\}, \] where \[ \left[ \begin\{matrix\} n\\ m\end\{matrix\} \right] =\prod \_\{k=1\}^m\frac\{1-q^\{n-k+1\}\}\{1-q^k\} \quad \text\{and\} \quad (a;q)\_n=\prod \_\{k=0\}^\{n-1\}(1-aq^k). \]},
author = {Liu, Ji-Cai, Zhao, Shan-Shan},
journal = {Czechoslovak Mathematical Journal},
keywords = {Gauss’ identity; $q$-binomial coefficient; $q$-binomial theorem},
language = {eng},
number = {4},
pages = {1183-1189},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Truncations of Gauss' square exponent theorem},
url = {http://eudml.org/doc/298941},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Liu, Ji-Cai
AU - Zhao, Shan-Shan
TI - Truncations of Gauss' square exponent theorem
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 1183
EP - 1189
AB - 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 $n$, \[ \sum _{k=0}^n(-1)^k \left[ \begin{matrix} 2n-k\\ k \end{matrix} \right] (q;q^2)_{n-k}q^{{k+1\atopwithdelims ()2}} =\sum _{k=-n}^n(-1)^kq^{k^2}, \] where \[ \left[ \begin{matrix} n\\ m\end{matrix} \right] =\prod _{k=1}^m\frac{1-q^{n-k+1}}{1-q^k} \quad \text{and} \quad (a;q)_n=\prod _{k=0}^{n-1}(1-aq^k). \]
LA - eng
KW - Gauss’ identity; $q$-binomial coefficient; $q$-binomial theorem
UR - http://eudml.org/doc/298941
ER -