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 ̸ = −2n 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.

Around q-Appell polynomial sequences

“…The polynomials having this property are known as Brenke-type polynomials. Although Chihara [5] explicitly determined all the orthogonal polynomial sequences generated by (11), it is not trivial to glean from that work the existence and subsequent deduction of all the orthogonal Appell sequence with respect to a given operator O. Notwithstanding this, we recognize this could be a procedure to adopt in order to deduce the same results that we will present here; but instead, our research will be based on an algebraic approach operating on the dual space P .…”

“…(with the convention: (xD) k+1 = xD (xD) k for any integer k ≥ 0) arisen from the quadratic decomposition of a Laguerre sequence, do not have a single MOPS being also G ε,μ -Appell [11].…”

Appell polynomial sequences with respect to some differential operators

“…As a last remark, we must add that a particular Λ operator with k = 3, so-called G ǫ,µ , was debated in [13]. It was proved the absence of orthogonal G ǫ,µ -Appell sequences indicating that Theorem 4.2 may be generalized at a latter time.…”

Generating Convergent Laguerre-Gaussian Beams Based on an Arrayed Convex Spiral Phaser Fabricated by 3D Printing

“…As known, vortex light beams demonstrate some potential applications such as quantum communication [ 1 , 2 , 3 , 4 , 5 ], trapping and manipulation of particles [ 6 , 7 , 8 , 9 ] and biomedicine [ 10 , 11 , 12 ], due to their unique orbital angular momentum [ 13 , 14 , 15 , 16 , 17 ]. In particular, the light fields of the Laguerre—Gaussian (LG) beams [ 18 , 19 , 20 , 21 ] contain the key Laguerre polynomials [ 22 , 23 , 24 , 25 ], which have a property of orthogonal normalization, so the LG beams can be used to form a complicated optical mode. Generally, any vortex beam can be considered as a linear combination of LG beams.…”