Symbolic approach to the general cubic decomposition of polynomial sequences. Results for several orthogonal and symmetric cases

Abstract

Abstract. We deal with a symbolic approach to the cubic decomposition (CD) of polynomial sequences -presented in a previous article referenced herein -which allows us to compute explicitly the first elements of the nine component sequences of a CD. Properties are investigated and several experimental results are discussed, related to the CD of some widely known orthogonal sequences. Results concerning the symmetric character of the component sequences are established.

On a 2-Orthogonal Polynomial Sequence via Quadratic Decomposition

“…In order to investigate the structure of each component we have gathered some short commands developed in [19], namely a generic command that computes the structure coefficients of any previously defined MPS, and further instructions that search for the possibility of defining a proper MPS from the secondary components.…”

“…In this paper, we decompose quadratically 2-orthogonal polynomial sequences, following the wider set up of the QD and we put in evidence a family for which the application of the symbolic implementation of the general QD presented in [11] together with broad symbolic procedures introduced in [19] provides interesting results. In particular, that 2-orthogonal family yields three further 2-orthogonal polynomial sequences through the most general quadratic decomposition.…”

Symbolic Approach to the General Quadratic Polynomial Decomposition

“…Let {W n } n≥0 be MOPS with constant recurrence coefficients, also considered in [31] in the context of cubic decomposition…”

“…Since then the type of QD has been generalized in order to decompose a nonsymmetric sequence [21,24] and to admit a full quadratic mapping leading to the so-called general quadratic decomposition (GQD) [16,17]. In this work we propose a symbolic approach to the GQD in the sequel of what was done in [31] for the general cubic decomposition [28].…”

Several characterizations of third degree semiclassical linear forms of class two appearing via cubic decomposition

“…Such constraints (symmetry and the particular choice of the cubic polynomials) are removed in [15,16], where the authors consider the problem of orthogonality of sequences {W n } n≥0 such that W 3n+m (x) = θ m (x)P n (π 3 (x)), m ∈ {0, 1, 2}, where π 3 (x) is a fixed cubic polynomial and θ m (x) is a fixed polynomial of degree m. The problem of general cubic decompositions of orthogonal polynomials is studied in [22]. Later on, in [23] a symbolic approach is presented. We should point out that, problems concerning cubic decompositions in the framework of the semiclassical orthogonal polynomial sequences have not been considered in the literature up to the recent contributions [27], [28] and [29], for particular cases of semiclassical and H q -semiclassical orthogonal polynomials of class one and two.…”