On irreducible p , q ‐representations of Lie algebras \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {G} (0,1)$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {G} (0,0)$\end{document}

In this paper, the irreducible p , q ‐representations of the Lie algebras \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {G}(0,1)$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {G}(0,0)$\end{document} are discussed. We prove two theorems that classify certain irreducible p , q ‐representations of these Lie algebras and construct their one variable models in terms of p , q ‐derivative and dilation operators. As an application, we derive a p , q ‐special function identity based on one such model.