Uniform Asymptotic Approximations for the Meixner–sobolev Polynomials

Abstract

We obtain uniform asymptotic approximations for the monic Meixner–Sobolev polynomials Sn(x). These approximations for n → ∞, are uniformly valid for x/n restricted to certain intervals, and are in terms of Airy functions. We also give asymptotic approximations for the location of the zeros of Sn(x), especially the small and the large zeros are discussed. As a limit case, we also give a new asymptotic approximation for the large zeros of the classical Meixner polynomials. The method is based on an integral representation in which a hypergeometric function appears in the integrand. After a transformation, the hypergeometric functions can be uniformly approximated by unity, and all that remains are simple integrals for which standard asymptotic methods are used. As far as we are aware, this is the first time that standard uniform asymptotic methods have been used for the Sobolev-class of orthogonal polynomials.

On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case

“…where β > 0, 0 < c < 1, and M, N ≥ 0. Other slightly more recent results, connected with the Sobolev-Meixner polynomials, can be found in [11,12]. Note that our work is in someways connected to the paper [13], where the authors consider the Charlier case.…”

“…In this section, we obtain a second-order linear difference equation that the sequence {M j, n (x; β, c; λ)} n≥0 satisfies. In order to achieve this, we will find the ladder (creation and annihilation) operators, using the connection Formula (26), the three-term recurrence relation (7), and the structure relations (10) and (11) satisfied by them.…”

New analytic properties of nonstandard Sobolev-type Charlier orthogonal polynomials

“…Concerning the norm of the polynomials Q λ n (x) we can state the following Theorem 1 Let Q λ n n≥0 be the sequence of Sobolev-type Charlier orthogonal polynomials defined by (17). Then, for every n ≥ 1, λ ∈ R + , c ∈ R such that ψ (a) has no points of increase in the interval (c, c + 1), and a > 0, the norm of these polynomials, orthogonal with respect to (1) is (15) we have…”

“…Since then, and to the best of our knowledge, the Charlier case has remained untouched. Several researchers have done further work on the Sobolev-type case for discrete orthogonality measures, but mainly concerning the Meixner case (see [4], [5], [15], [20], [21] and the references given there).…”

Asymptotic expansions of the hypergeometric function with two large parameters—application to the partition function of a lattice gas in a field of traps

“…Furthermore, the conditions on the parameters when using the transformation formulae of the HGF strongly restrict the expansions obtained after applying the transformations. After some advances in [2] and [3] for a particular HGF occurring in fluid flow theory, the most exhaustive treatment of the cases ε i ∈ {0, 1, −1} in (5) was provided recently by Olde Daalhuis [39][40][41][42][43][44] and Jones [45]. The resulting AEs took the form of series of Airy [40,41,43], parabolic cylinder [39,41,43], Bessel [43,45], Hankel [43], and/or Kummer [42][43][44] functions, depending on the value of ε i .…”