Quasi-orthogonality of some hypergeometric polynomials

Abstract

In this paper, we prove the quasi-orthogonality of a family of 2 F 2 polynomials and several classes of 3 F 2 polynomials that do not appear in the Askey scheme for hypergeometric orthogonal polynomials. Our results include, as a special case, two 3 F 2 polynomials considered by Dickinson in 1961. We also discuss the location and interlacing of the real zeros of our polynomials.

Interlacing of zeros of quasi-orthogonal Meixner polynomials

“…Quasi-orthogonality has been investigated by many authors, including Fejér [9], Shohat [22], Chihara [4], Dickinson [6], Draux [7], Maroni [17] and Joulak [15]. The quasi-orthogonality of Jacobi, Gegenbauer and Laguerre sequences is discussed in [2], and the quasi-orthogonality of Meixner sequences in [13] and of Meixner-Pollaczek, Hahn, Dual-Hahn and Continuous Dual-Hahn sequences in [11]. Recently, in [3], interlacing properties of zeros of quasiorthogonal polynomials were used to prove results on Gaussian-type quadrature.…”

On Certain Properties and Applications of the Perturbed Meixner–Pollaczek Weight

“…(This fact is later checked with numerical experiments of the zeros of these polynomials). The quasi-orthogonality of the monic Meixner-Pollaczek polynomials is therefore of even order, as detailed in the next result ( [17], Theorem 3.3).…”

Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials

“…In this section we consider the quasi-orthogonality of the monic Wilson and Racah polynomials, that are defined on a quadratic lattice. The quasi-orthogonality of the dual Hahn and continuous dual Hahn polynomials, that also fall in this category, was discussed in [15]. We also prove the quasi-orthogonality of the monic continuous Hahn polynomials that are defined on a linear lattice.…”

“…Quasi-orthogonality was first studied by Riesz [25], followed by Fejér [12], Shohat [26], Chihara [4], Dickinson [6], Draux [7], Maroni [22] and Joulak [17]. The quasi-orthogonality of Jacobi, Gegenbauer and Laguerre sequences is discussed in [1], the quasi-orthogonality of Meixner sequences in [16] and of Meixner-Pollaczek, Hahn, dual Hahn and continuous dual Hahn sequences in [15]. More recently, interlacing of zeros of quasi-orthogonal Meixner, Jacobi, Laguerre and Gegenbauer polynomials were studied in [8,9,10,11] and in [2] interlacing properties of zeros of quasi-orthogonal polynomials were used to prove results on Gaussian-type quadrature.…”