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.
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