On the zeros of Meixner polynomials

Abstract

We investigate the zeros of a family of hypergeometric polynomials

Interlacing of zeros of quasi-orthogonal Meixner polynomials

“…It is known, [13], that as the parameter β decreases below zero, the zeros of the Meixner polynomial M n (x; β, c) depart from the interval of orthogonality (0, ∞) through the origin. In particular, for each fixed β and c, −β, c ∈ (0, 1) one zero of M n (x; β, c) is simple and negative, the remaining (n − 1) zeros are simple and positive and the Meixner sequence M n (x; β, c)} ∞ n=0 is quasi-orthogonal of order 1.…”

“…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.…”

Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials

“…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.…”

Meixner Polinomlarının Bazı Yeni Özellikleri

“…with similar interpretations for other sets of parameters (see [4], [10], [14], [20]). Here, as usual,  ) ( denotes the Pochhammer symbol and Lauricella functions theory and its applications (see [17], [18], [20]).…”

“…In addition, we derive a theorem giving certain families of bilateral generating functions for the generalized Lauricella functions and the Meixner polynomials. Nowadays, there are a lot of works related to Meixner polynomials and Lauricella functions theory and its applications (see [17], [18], [20]). Lemma 1.1.…”