On models of irreducible $q$-representations of Lie algebra ${\mathcal{𝒦}}_5$

In this paper, we construct models of irreducible $q$-representations of the five-dimensional Lie algebra ${\mathcal{𝒦}}_5$ in terms of $q$-derivative and $q$-dilation operators. These models are transformed to new models of ${\mathcal{𝒦}}_5$ using the theory of fractional $q$-calculus and $q$-Euler integral transformation. The transformed models are in terms of inverse $q$-derivative operators and difference $q$-dilation operators. These models are further exploited to obtain interesting recurrence relations and matrix elements.