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In 2000, Ariki and Mathas showed that the simple modules of the Ariki--Koikealgebras $\mathcal{H}_{\mathbb{C},q;Q_1,\ldots, Q_m}\big(G(m, 1, n)\big)$ (whenthe parameters are roots of unity and $q\neq 1$) are labeled by the so-calledKleshchev multipartitions. This together with Ariki's categorification theoremenabled Ariki and Mathas to obtain the generating function for the number ofKleshchev multipartitions by making use of the Weyl--Kac character formula. Inthis paper, we revisit this generating function for the $q=-1$ case. This$q=-1$ case is particularly interesting, for the corresponding Kleshchevmultipartitions have a very close connection to generalized Rogers--Ramanujantype partitions when $Q_1=\cdots=Q_a=-1$ and $Q_{a+1}=\cdots =Q_m =1$. Based onthis connection, we provide an analytic proof of the result of Ariki and Mathasfor $q=Q_1=\cdots Q_a=-1$ and $Q_{a+1}=\cdots =Q_m =1$. Our second objective isto investigate simple modules of the Ariki--Koike algebra in a fixed block. Itis known that these simple modules in a fixed block are labeled by theKleshchev multiparitions with a fixed partition residue statistic. Thispartition statistic is also studied in the works of Berkovich, Garvan, andUncu. Employing their results, we provide two bivariate generating functionidentities when $m=2$.

The Ariki--Koike algebras and Rogers--Ramanujan type partitions