Uniform Asymptotics of the Meixner Polynomials

Fe b 20 09 Uniform Asymptotics of the Meixner Polynomials

Using the steepest descent method of Deift-Zhou, we derive locally uniform asymptotic formulas for the Meixner polynomials. These include an asymptotic formula in a neighborhood of the origin, a result which as far as we are aware has not yet been obtained previously. This particular formula involves a Cauchy integral, which is the uniformly bounded solution to a scalar Riemann-Hilbert problem,...

Global asymptotics of the Meixner polynomials

Asymptotics of Derivatives of Orthogonal Polynomials on the Real Line

We show that uniform asymptotics of orthogonal polynomials on the real line imply uniform asymptotics for all their derivatives. This is more technically challenging than the corresponding problem on the unit circle. We also examine asymptotics in the L2 norm. 1. Results Let μ be a nite positive Borel measure on [−1, 1] and let {pn}n=0 denote the corresponding orthonormal polynomials, so that ∫...

Weighted Derangements and the Linearization Coefficients of Orthogonal Sheffer Polynomials

The present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal Sheffer polynomials, i.e., Hermite, Charlier, Laguerre, Meixner and Meixner-Pollaczek polynomials. In particular, we show that Viennot's combinatorial interpretations of the moments can be derived directly from their classical analytical exp...

Bounds for zeros of Meixner and Kravchuk polynomials

The zeros of certain different sequences of orthogonal polynomials interlace in a well-defined way. The study of this phenomenon and the conditions under which it holds lead to a set of points that can be applied as bounds for the extreme zeros of the polynomials. We consider different sequences of the discrete orthogonal Meixner and Kravchuk polynomials and use mixed three term recurrence rela...