A new q-analogue of the binomial identity \sum_{k}(-۱)^k{۲n\choose n+۳k}=۲\cdot ۳^{n-۱}

In this paper, we establish a new q-analogue of the binomial identity:\begin{align*}&\sum_{k}(-۱)^k{۲n\choose n+۳k}=\begin{cases}۱,&\text{if n=۰,}\\[۵pt]۲\cdot۳^{n-۱},&\text{if n\ge ۱.}\end{cases}\end{align*}Our proof relies on a weight-preserving and sign-reversing involution due to Guo and Zhang.

Keywords:

q-binomial coefficient , q-binomial theorem , involution