Quadratic decomposition of Laguerre polynomials via lowering operators

A Laguerre polynomial sequence of parameter ε/2ε/2 was previously characterized in a recent work [Ana F. Loureiro and P. Maroni (2008) [28]] as an orthogonal FεFε-Appell sequence, where FεFε represents a lowering (or annihilating) operator depending on the complex parameter ε≠−2nε≠−2n for any integer n⩾0n⩾0. Here, we proceed to the quadratic decomposition of an FεFε-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 Gε,μGε,μ, where μμ is either 11 or −1−1. Therefore, we introduce and develop the concept of the Gε,μGε,μ-Appell sequences and we prove that they cannot be orthogonal. Finally, the quadratic decomposition of the non-symmetric sequence of Laguerre polynomials (with parameter ε/2ε/2) is fully accomplished.