Combinatorial Proofs of a Kind of Binomial and \(q\)-Binomial Coefficient Identities

Abstract

We give combinatorial proofs of some binomial and $q$-binomial identities in the literature, such as

\[\sum\limits_{k={-\infty}}^{\infty}(-1)^kq^{\frac{(9k^2+3k)}{2}}\binom{2n}{n+3k}=(1+q^n)\prod\limits_{k=1}^{n-1}(1+q^k+q^{2k})(n\geq 1)\]

and

\[\sum\limits_{k=0}^{\infty} \binom{3n}{2k}(-3)^k=(-8)^n.\]

Two related conjectures are proposed at the end of this paper.