Quadratic decomposition of Laguerre polynomials via lowering operators

Abstract

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 F ε -Appell sequence, where F ε represents a lowering (or annihilating) operator depending on the complex parameter ε ≠ − 2 n for any integer n ⩾ 0 . Here, we proceed to the quadratic decomposition of an F ε -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 ε , μ , where μ is either 1 or − 1 . Therefore, we introduce and develop the concept of the G ε , 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.