Uniform asymptotics for Jacobi polynomials with varying large negative parameters— a Riemann-Hilbert approach

2008

Numer Algor

Self Cite

By using the steepest descent method for Riemann-Hilbert problems introduced by Deift-Zhou (Ann Math 137:295-370, 1993), we derive two asymptotic expansions for the scaled Laguerre polynomial L (α) n (νz) as n → ∞, where ν = 4n + 2α + 2. One expansion holds uniformly in a right halfplane Re z ≥ δ 1 , 0 < δ 1 < 1, which contains the critical point z = 1; the other expansion holds uniformly in a left half-plane Re z ≤ 1 − δ 2 , 0 < δ 2 < 1 − δ 1 , which contains the other critical point z = 0. The two half-planes together cover the entire complex z-plane. The critical points z = 1 and z = 0 correspond, respectively, to the turning point and the singularity of the differential equation satisfied by L (α) n (νz).

Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach

Qiu

2008

Numer Algor

Self Cite

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

Dai

2008

Ramanujan J

Self Cite

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*

Dai

Chin. Ann. Math. Ser. B

2007

In this paper, we study the asymptotics of the Krawtchouk polynomials K N n (z; p, q) as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters n/N tends to a limit c ∈ (0, 1) as n → ∞. The results are globally valid in one or two regions in the complex z-plane depending on the values of c and p; in particular, they are valid in regions containing the interval on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou.2000 Mathematics Subject Classification. Primary 41A60, 33C45.