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The multiple zeta values (MZVs) have been studied extensively in recentyears. Currently there exist a few different types of $q$-analogs of the MZVs($q$-MZVs) defined and studied by mathematicians and physicists. In this paper,we give a uniform treatment of these $q$-MZVs by considering their doubleshuffle relations (DBSFs) and duality relations. The main idea is amodification and generalization of the one used by Castillo Medina et al. whohave considered the DBSFs of a special type of $q$-MZVs. We generalize theirmethod to a few other types of $q$-MZVs including the one defined by the authorin 2003. With different approach, Takeyama has already studied this type by"regularization" and observed that there exist $\mathbb Q$-linear relationswhich are not consequences of the DBSFs. He also discovered a new family ofrelations which we call the duality relations in this paper. This deficiency ofDBSFs occurs among other types, too, so we generalize the duality relations toall of these values and find that there are still some missing relations. Thisleads to the most general type of $q$-MZVs together with a new kind ofrelations called $\bf P$-$\bf Q$ relations which are used to lower thedeficiencies further. As an application, we will confirm a conjecture ofOkounkov on the dimensions of certain $q$-MZV spaces, either theoretically ornumerically, for the weight up to 12. Some relevant numerical data are providedat the end.

Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras