Around q-Appell polynomial sequences

Abstract

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 e 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 qe/2+1 are achieved, up to a linear transformation. This brings a new characterization of these polynomial sequences.