On the Asymptotics of the Meixner—Pollaczek Polynomials and Their Zeros

A Uniform Asymptotic Expansion for Krawtchouk Polynomials

“…1a 1d, one readily sees that the movements of w + and w & are very similar to those studied in [6] for the Meixner polynomials and in [8] for the Meixner Pollaczek polynomials. …”

“…Since the phase function F(w, *) in (2.6) has two saddle points w + (*) and w & (*) and these two points coalesce when *=* + and *=* & , our present situation is very much like those in the cases of Meixner [6] and Meixner Pollaczek [8] polynomials. Thus, we should compare Krawtchouk polynomials with the parabolic cylinder function given by…”

“…Case In all cases (i) (v), we have found that either Re f=0 and Im f is strictly increasing or Im f=0 and Re f is strictly increasing. Furthermore, the range of & With the values of ; and # so chosen, the transformation w W z defined in (4.6) can be shown, as in many previous papers [2,3,6,8], to be one-to-one and analytic along the whole steepest descent path through relevant saddle points of the integral in (2.5). The shapes of this steepest descent path, denoted by 1, and their images in the z-plane, denoted by 1 $, are depicted in Figs.…”

“…(Here we have used the fact that # is real; this can be proved as in [8].) The asymptotic approximation (6.14) can now be written as…”

Linear difference equations with transition points

“…However, so far this new method has not been able to produce results as strong as those obtainable from the classical approaches when an integral representation is available or when the differential equation theory can be applied. For instance, in the case of MeixnerPollaczek polynomials M n (x; δ, η), one can use a Cauchy integral representation to derive an infinite asymptotic expansion for M n (αn; δ, η), which holds uniformly for −M ≤ α ≤ M , where M can be any positive number; see [14]. Also, when the polynomial Q(x) = x 2m + · · · in the weight function w(x) = e −Q(x) is even and convex, one can use the turning point theory for differential equations to obtain an asymptotic formula for the polynomials p n (x) orthogonal with respect to w(x), which holds uniformly in the unbounded interval 0 ≤ x ≤ O(n 1/2m ); see [16].…”