$q$-de Rham complexes of higher level

In this article, we construct two kinds of de Rham-like complexes whichcompute the cohomology of complete crystals on higher-level $q$-crystallinesite, which was introduced in the previous article of the author. One complexis the $q$-analog of the higher de Rham complex constructed by Miyatani, andanother complex is the $q$-analog of the jet complex constructed by LeStum-Quir\'os. The complexes we constructed can also be regarded as thehigher-level analogs of the $q$-de Rham complex.