Quadratic decomposition of Laguerre polynomials via lowering operators

A Laguerre polynomial sequence of parameter ε/2 was previously characterized in a recent work [Ana F. Loureiro and P. Maroni (2008) [28]] as an orthogonal Fε-Appell sequence, where Fε represents a lowering (or annihilating) operator depending on the complex parameter ε ≠ −2n for any integer n ⩾ 0. Here, we proceed to the quadratic decomposition of anFε-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,μ, where μ is either 1 or −1. Therefore, we introduce and develop the concept of the Gε,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/2) is fully accomplished. c ⃝ 2010 Elsevier Inc. All rights reserved.