Let \(\mu\geq-1/2\) and let \(\{M_p\}_{p=0}^\infty\) be a sequence of continuous functions defined on \(I=(0,\infty)\) such that \[ 1=M_0(x)\leq M_1(x)\leq M_2(x)\leq\dots. \] Then \(K_\mu\{M_p\}\) is a Hankel-\(K\{M_p\}\) space of order \(\mu\) if it consists of all functions \(\varphi\in C^\infty(I)\) such that \[ | | \varphi| | _{\mu,p} =\max_{0\leq k\leq p} \sup_{x\in I}\left| M_p(x) (x^{-1}D)^k x^{-\mu-1/2}\varphi(x)\right| <\infty\quad (p\in \mathbb N). \] By \(K_\mu\{M_p\}'\) the dual space of \(K_\mu\{M_p\}\) is denoted. In this paper, the author gives characterizations of membership, boundedness and convergence in \(K_\mu\{M_p\}'\), using the same technique as A. Kaminski [Stud. Math. 77, 499–508 (1984; Zbl 0547.46019)].