Quadratic decomposition of Appell sequences

We proceed to the quadratic decomposition of Appell sequences and we characterise the four derived sequences obtained by this approach. We prove that the two monic polynomial sequences associated to such quadratic decomposition are also Appell sequences with respect to another (lowering) operator, which we call as F?, where either ?=1 or -1. Thus, we introduce and develop the concept of the Appell polynomial sequences with respect to the operator F? (where, ? is a parameter belonging to the field of the complex numbers): the F?-Appell sequences. The orthogonal polynomial sequences that are also F?-Appell correspond to the Laguerre sequences with parameter ?/2. Indeed, this brings an entirely new characterisation of the Laguerre sequences.