On the $q$-Ces\`{a}ro bounded double sequence space

In this article, the new sequence space $\tilde{\mathcal{M}}_u^q$ is acquainted, described as the domain of the 4d (4-dimensional) $q$-Ces\`{a}ro matrix operator, which is the $q$-analogue of the first order 4d Ces\`{a}ro matrix operator, on the space of bounded sequences. In the continuation of the study, the completeness of the new space is given, its fundamental set is found, and the inclusion relation related to the space is presented. In the last two parts, the duals of the space are determined and some matrix classes are acquired.