Global Asymptotics of Stieltjes–Wigert Polynomials (original) (raw)

Asymptotics and zero distribution of Padé polynomials associated with the exponential function

Journal of Computational and Applied Mathematics, 1998

The polynomials P n and Q m having degrees n and m respectively, with P n monic, that solve the approximation problem P n (z)e z + Q m (z) = O z n + m +1 will be investigated for their asymptotic behavior, in particular in connection with the distribution of their zeros. The symbol O means that the left-hand side should vanish at the origin at least to the order n+m+ 1. This problem is discussed in great detail in a series of papers by Sa and Varga. In the present paper we show h o w their results can be obtained by using uniform expansions of integrals in which Airy functions are the main approximants. We give approximations of the zeros of P n and Q m in terms of zeros of certain Airy functions, as well of those of the remainder dened by E n;m (z) = P n (z) e z + Q m (z) .

Asymptotic Properties of Heine-Stieltjes and Van Vleck Polynomials

Journal of Approximation Theory, 2002

We study the asymptotic behavior of the zeros of polynomial solutions of a class of generalized Lam! e e differential equations, when their coefficients satisfy certain asymptotic conditions. The limit distribution is described by an equilibrium measure in the presence of an external field, generated by charges at the singular points of the equation. Moreover, a case of non-positive charges is considered, which leads to an equilibrium with a non-convex external field. # 2002 Elsevier Science (USA)

The Role of Hermite Polynomials in Asymptotic Analysis

Special Functions - Proceedings of the International Workshop, 2000

Hermite polynomials are considered as appraximants in asymptotic representations of certain other polynomials. Examples are given for polynomials from the Askey scheme of hypergeometric orthogonal polynomials. We also mention that Hermite polynomials can be used as main approximants in uniform asymptotic representations of certain types of integrals and differential equations.

Asymptotics of orthogonal polynomials inside the unit circle and Szegő–Padé approximants

Journal of Computational and Applied Mathematics, 2001

We study the asymptotic behavior of orthogonal polynomials inside the unit circle for a subclass of measures that satisfy Szegő's condition. We give a connection between such behavior and a Montessus de Ballore-type theorem for Szegő-Padà e rational approximants of the corresponding Szegő function.

Asymptotics and exact formulas for Zagier polynomials

Research in Number Theory, 2016

In 1998 Don Zagier introduced the modified Bernoulli numbers B * n and showed that they satisfy amusing variants of some properties of Bernoulli numbers. In particular, he studied the asymptotic behavior of B * 2n , and also obtained an exact formula for them, the motivation for which came from the representation of B 2n in terms of the Riemann zeta function ζ(2n). The modified Bernoulli numbers were recently generalized to Zagier polynomials B * n (x). For 0 < x < 1, an exact formula for B * 2n (x) involving infinite series of Bessel function of the second kind and Chebyshev polynomials, that yields Zagier's formula in a limiting case, is established here. Such series arise in diffraction theory. An analogous formula for B * 2n+1 (x) is also presented. The 6periodicity of B * 2n+1 is deduced as a limiting case of it. These formulas are reminiscent of the Fourier expansions of Bernoulli polynomials. Some new results, for example, the one yielding the derivative of the Bessel function of the first kind with respect to its order as the Fourier coefficient of a function involving Chebyshev polynomials, are obtained in the course of proving these exact formulas. The asymptotic behavior of Zagier polynomials is also derived from them. Finally, a Zagier-type exact formula is obtained for B * 2n − 3 2 + B * 2n .

Strong asymptotics of multi-level Hermite-Pad\'e polynomials

Cornell University - arXiv, 2022

We obtain the strong asymptotics of multiple orthogonal polynomials which arise in a mixed type Hermite-Padé approximation problem defined on a Nikishin system of functions. The results obtained allow to give exact estimates of the rate of convergence of the approximating rational functions and the strong asymptotics of Cauchy biorthogonal polynomials.