Global asymptotics for Laguerre polynomials with large negative parameter—a Riemann-Hilbert approach

Abstract

In this paper, we study the asymptotic behavior of the Laguerre polynomials L (α n ) n (nz) as n → ∞. Here α n is a sequence of negative numbers and −α n /n tends to a limit A > 1 as n → ∞. An asymptotic expansion is obtained, which is uniformly valid in the upper half plane C + = {z : Im z ≥ 0}. A corresponding expansion is also given for the lower half plane C − = {z : Im z ≤ 0}. The two expansions hold, in particular, in regions containing the curve in the complex plane, on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou.

Global Asymptotics of Krawtchouk Polynomials – a Riemann-Hilbert Approach*

“…After transforming the discrete RHP associated with this polynomial into a specific continuous one, we find that this problem is similar to some of the problems considered previously (see, e.g., [23], [24] and [26]), and our method in [5] can be applied. More precisely, for 0 < c < p, we present an infinite asymptotic expansion which is valid uniformly in the whole complex plane bounded away from (−∞, 0] and [1, ∞).…”

“…Furthermore, by successive approximation (see [5]), it can be easily demonstrated that the expansion (3.59) holds uniformly for all…”

“…If we can transform a discrete Riemann-Hilbert problem into a continuous one, then we can apply the techniques that we have developed in [5] and [23]. Due to the sensitivity of parameter c, this transformation is different in different cases.…”

“…The advantage of adopting the parabolic cylinder function, over the Airy function as done in [5], is that the region of validity of the asymptotic expansion we get here can include both the critical values α n and β n , whereas the region of validity of the Airy-type expansion given in [5] includes only one critical value.…”

“…With a similar technique as given in [5], we construct the parametrix of the RHP for V * by using the Airy functions in these four regions. Define …”