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

$$\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^k{q}^{\frac{(9k^2+3k)}{2}}(\genfrac{}{}{0ex}{}{2n}{n+3k})=(1+q^n)\prod _{k=1}^{n-1}(1+q^k+q^{2k})(n\ge 1)$$

and

$$\sum _{k=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{3n}{2k})(-3{)}^k=(-8{)}^n.$$

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