  Recently, $4$-regular partitions into distinct parts are connected with a
family of overpartitions. In this paper, we provide a uniform extension of two
relations due to Andrews for the two types of partitions. Such an extension is
made possible with recourse to a new trivariate Rogers--Ramanujan type
identity, which concerns a family of quadruple summations appearing as
generating functions for the aforementioned overpartitions. More interestingly,
the derivation of this Rogers--Ramanujan type identity is relevant to a certain
well-poised basic hypergeometric series.
