Narrow Results

In this paper, we investigate the asymptotic behavior of the generalized Bessel polynomials Y_n(z; a). Let z = α/(n + 1). We first derive infinite asymptotic expansions for y_n(z; a) when α lies in various regions of the complex plane, except when α is near ± i. Then we construct uniform asymptotic expansions for Y_n(z; a) in neighborhoods of α = ± i. These expansions involve the Airy function and its derivative. Finally, we use these approximations to study the asymptotic behavior of the zeros of Y_n(z; a) near α = i.

Let j_v,k denote the k-th positive zero of the Bessel function j_v,k. In this paper, we prove that for í > 0 and k = 1, 2, 3, … ,

Let {p_n(x)}n≥0 be the set of orthonormal polynomials with respect to the exponential weight w(x) = e^−υ(x), where υ(x) = x^2m + … is a monic polynomial of degree 2m with m ≥ 2 and is even. An asymptotic approximation is obtained for p_n(x), as n→∞, which holds uniformly for 0 ≤ x ∞ O(n^½m). As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.

An infinite asymptotic expansion is derived for the Meixner-Pollaczek polynomials M_n (n_α; δ, η) as n → ∞, which holds uniformly for −M ≤ α ≤ α M, where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α_n,s denotes the sth zero of Mn (n_α; δ, η), counted from the right, and if , denotes its sth zero counted from the left, then for each fxed s, three-term asymptotic approximations are obtained for both α_n,s and , as n → ∞.

It has been known for some time that the existing asymptotic methods for integrals and differential equations are not applicable in the case of Stieltjes–Wigert polynomials with degree going to infinity. Using the recently introduced nonlinear steepest descent method for Riemann–Hilbert problems, here we not only derive an asymptotic expansion for these polynomials, but we also show that the result holds uniformly in the complex plane except for a sector containing the real axis from −∞ to 1/4. Furthermore, we give an asymptotic formula for the zeros of these polynomials, which approximates the true values of the zeros closely.

On sait depuis longtemps que les méthodes asymptotiques existantes pour les intégrales et les équations différentielles ne sont pas applicables aux polynômes de Stieltjes–Wigert lorsque le degré tend vers l'infini. En utilisant des méthodes de descente non lineaires, introduites récemment pour les problèmes de Riemann–Hilbert, nous obtenons non seulement un développement asymptotique pour ces polynômes, mais nous montrons également que le résultat est uniforme dans le plan complexe, sauf pour un secteur contenant l'axe réel de −∞ à 1/4. De plus, nous démontrons une formule asymptotique qui donne une bonne approximation des racines de ces polynômes.