Around q-Appell polynomial sequences

First we show that the quadratic decomposition of the Appell polynomials with respect to the q-divided difference operator is supplied by two other Appell sequences with respect to a new operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}_{q;q^{-\varepsilon}}$\end{document}, where ε represents a complex parameter different from any negative even integer number. While seeking all the orthogonal polynomial sequences invariant under the action of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}_{\sqrt{q};q^{-\varepsilon/2}}$\end{document} (the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{M}_{\sqrt{q};q^{-\varepsilon/2}}$\end{document}-Appell), only the Wall q-polynomials with parameter qε/2+1 are achieved, up to a linear transformation. This brings a new characterization of these polynomial sequences.