A new $q$-analogue of the binomial identity $\sum_{k}(-1)^k{2n\choose n+3k}=2\cdot 3^{n-1}$
نویسندگان
1 Department of Mathematics, Wenzhou University, Wenzhou, PR China
2 Department of Mathematics, Wenzhou University, Wenzhou, PR China
doi
10.22108/toc.2023.135277.2017چکیده
In this paper, we establish a new $q$-analogue of the binomial identity:\begin{align*}&\sum_{k}(-1)^k{2n\choose n+3k}=\begin{cases}1,&\text{if $n=0$,}\\[5pt]2\cdot3^{n-1},&\text{if $n\ge 1$.}\end{cases}\end{align*}Our proof relies on a weight-preserving and sign-reversing involution due to Guo and Zhang.
