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 (mathcal{M}_{q;q^{-varepsilon}}), 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 (mathcal{M}_{sqrt{q};q^{-varepsilon/2}}) (the (mathcal{M}_{sqrt{q};q^{-varepsilon/2}})-Appell), only the Wall q-polynomials with parameter q ^ε/2+1 are achieved, up to a linear transformation. This brings a new characterization of these polynomial sequences.