An inverse problem associated with (1, 1) symmetric coherent linear functionals. Issue 7 (3rd July 2019)

• Record Type: Journal Article
• Authors: Ruiz, Herbert Dueñas; Español, Francisco Marcellán; Molano, Alejandro Molano
• Is Part Of: Integral transforms and special functions. Volume 30:Issue 7(2019)
• Page Start: 526
• Page End: 546
• Publication Date: 2019-07-03
• Subjects: Orthogonal polynomials -- inverse problems -- semiclassical linear functionals; 33C45 -- 42C05; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2019.1590354
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: In this paper we study an inverse problem associated with the non-coherent algebraic relation P n + 1 [ i ] ( x ) + a n [ 1 ] P n [ i ] ( x ) + a n [ 2 ] P n − 1 [ i ] ( x ) + b n Q n + 1 ( x ) + c n Q n ( x ) = 1 + b n R n + 1 ( x ) + d n R n ( x ), n ≥ 0, where the sequences of monic polynomials P n ( x ) n ≥ 0, Q n ( x ) n ≥ 0 and R n ( x ) n ≥ 0 are orthogonal with respect to quasi-definite linear functionals u, v and w, respectively, and P k [ i ] ( x ) = P k + i ( i ) ( x ) ( k + 1 ) i for i = 0, 1. We assume that v is a monic polynomial perturbation of u . Moreover, in the case i = 1, we assume that u is a semiclassical linear functional of class s . In this way, we discuss the relation between the formal Stieltjes series associated with u and w . This inverse problem is motivated by the analysis of sequences of polynomials orthogonal with respect to ( 1, 1 ) symmetric coherent pairs of linear functionals when a symmetrization process is implemented.