Quadratic decomposition of Laguerre polynomials via lowering operators

A Laguerre polynomial sequence of parameter ?/2 was previously characterized in a recent work [Ana F. Loureiro and P. Maroni (2008) [28]] as an orthogonal F?-Appell sequence, where F? represents a lowering (or annihilating) operator depending on the complex parameter ???2n for any integer n?0. Here, we proceed to the quadratic decomposition of an F?-Appell sequence, and we conclude that the four sequences obtained by this approach are also Appell but with respect to another lowering operator consisting of a Fourth-order linear differential operator G?,?, where ? is either 1 or ?1. Therefore, we introduce and develop the concept of the G?,?-Appell sequences and we prove that they cannot be orthogonal. Finally, the quadratic decomposition of the non-symmetric sequence of Laguerre polynomials (with parameter ?/2) is fully accomplished.