Weight $q$-multiplicities for representations of $\mathfrak{sp}_4(\mathbb{C})$

In this paper we present a closed formula for the values of the $q$-analog of Kostant's partition function for the Lie algebra $\mathfrak{sp}_4(\mathbb{C})$ and use this result to give a simple formula for the $q$-multiplicity of a weight in the representations of the Lie algebra $\mathfrak{sp}_4(\mathbb{C})$. This generalizes the 2012 work of Refaghat and Shahryari that presented a closed formula for weight multiplicities in representations of the Lie algebra $\mathfrak{sp}_4(\mathbb{C})$.