Asymptotics for the zeros of the generalized Bessel polynomials

Abstract

Summary. We investigate the location of the zeros of the normalized generalized Bessel polynomials and the normalized reversed generalized Bessel polynomials. Also, the rate at which these zeros approach certain well-defined curves is investigated. On the basis of numerical computations and graphs, four new conjectures are proposed.

Riemann-Hilbert Analysis for Laguerre Polynomials with Large Negative Parameter

“…Stated in terms of generalized Bessel polynomials (1.11)-(1.12) asymptotic results on zeros are due to De Bruin et al [4], Carpenter [5], and Wong and Zhang [31]. These papers deal with the limit (1.13) with A = 2.…”

Asymptotics for the zeros and poles of normalized Pad\'{e} approximants to ${\rm e}^{z}$

“…1, are also measured, as in Corollary 2, against a fixed curve, D 0,∞ , but we note with interest that the result of (2.6) for 0 < σ < ∞, which is the analog of (1.6), now eliminates the (ln n) term appearing in (1.6). We also remark that essentially the special case σ = 1 of Corollary 2 is obtained (via a different technique) in [3]. To illustrate the results of Theorem 1 and Corollary 2, we have graphed in Fig.…”

“…(9.31)] that the negative real zero z nj ,νj of R nj ,νj ((n j + ν j )z) satisfies z nj ,νj =ẑ nj,νj + O 1 (n j + ν j ) 2 (j → ∞), (2.11) whereẑ nj ,νj denotes the real point of the arc D σj ,nj +νj . We also remark that the result of Theorem 3, for essentially the case σ j = 1, is obtained in [3]. To illustrate the result of Theorem 3, we have graphed in Figs.…”

Accurate computation of the zeros of the generalized Bessel polynomials

“…Also, let n ≥ 1. Then Remark 3.4 As far as we know, no result better than those in the above theorems has been obtained about the location of the zeros of GBPs, except for results in [10] and [33] which, however, are valid only asymptotically for n → ∞.…”

“…Remark 3.2 Formula (3.5) shows that the value of the constant b is of little importance in the study of the u (a,b) n 's. Almost all authors (see, e.g., [2,4,10,12,25,26,32,33] and [44]) assume b = 2. . It follows from Theorem 3.1 (and also from (3.5)) that d is not always equal to the parameter n in (3.1) and that one has d = n if and only if…”