A $q$-series identity via the $\mathfrak{sl}_3$ colored Jones polynomials for the $(2,2m)$-torus link

The colored Jones polynomial is a $q$-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A $q$-series called a tail is obtained as the limit of the $\mathfrak{sl}_2$ colored Jones polynomials $\{J_n(K;q)\}_n$ for some link $K$, for example, an alternating link. For the $\mathfrak{sl}_3$ colored Jones polynomials, the existence of a tail is unknown. We give two explicit formulas of the tail of the $\mathfrak{sl}_3$ colored Jones polynomials colored by $(n,0)$ for the $(2,2m)$-torus link. These two expressions of the tail provide an identity of $q$-series. This is a knot-theoretical generalization of the Andrews-Gordon identities for the Ramanujan false theta function.