The power collection method for connection relations: Meixner polynomials

Uniform Asymptotics of the Meixner Polynomials

Using the Deift-Zhou steepest descent method, we derive locally uniform asymptotic formulas for the Meixner polynomials. So far as we know, the asymptotic behavior in a neighborhood of the origin has not been studied before. To fill this gap, we should impose a Cauchy integral, which is the uniformly bounded solution to a scalar Riemann-Hilbert problem, and which converges exponentially fast (a...

On the zeros of Meixner polynomials

We investigate the zeros of a family of hypergeometric polynomials Mn(x;β, c) = (β)n 2F1(−n,−x;β; 1 − 1c ), n ∈ N, known as Meixner polynomials, that are orthogonal on (0,∞) with respect to a discrete measure for β > 0 and 0 < c < 1. When β = −N , N ∈ N and c = p p−1 , the polynomials Kn(x; p,N) = (−N)n 2F1(−n,−x;−N ; 1 p ), n = 0, 1, . . . N , 0 < p < 1 are referred to as Krawtchouk polynomial...

Meixner Polynomials and Random Partitions

The paper deals with a 3-parameter family of probability measures on the set of partitions, called the z-measures. The z-measures first emerged in connection with the problem of harmonic analysis on the infinite symmetric group. They are a special and distinguished case of Okounkov’s Schur measures. It is known that any Schur measure determines a determinantal point process on the 1-dimensional...

Global asymptotics of the Meixner polynomials

The indeterminate moment problem for the q-Meixner polynomials

For a class of orthogonal polynomials related to the q-Meixner polynomials corresponding to an indeterminate moment problem we give a one-parameter family of orthogonality measures. For these measures we complement the orthogonal polynomials to an orthogonal basis for the corresponding weighted L-space explicitly. The result is proved in two ways; by a spectral decomposition of a suitable opera...