The Riemann–hilbert Approach to Global Asymptotics of Discrete Orthogonal Polynomials With Infinite Nodes

Asymptotics of the Charlier polynomials via difference equation methods

“…For discrete orthogonal polynomials, pioneering work has been done by Baik et al [2], published in 2007. The method of Baik et al is applied by Ou and Wong [18], with modifications, to obtain uniform asymptotic approximations for C (a) n (z) with non-rescaled complex variable z. A significant feature of [18] is the global uniformity.…”

“…The method of Baik et al is applied by Ou and Wong [18], with modifications, to obtain uniform asymptotic approximations for C (a) n (z) with non-rescaled complex variable z. A significant feature of [18] is the global uniformity. Asymptotic expansions are derived in three regions that cover the whole complex plane.…”

Uniform asymptotics for the discrete Laguerre polynomials

“…For the case c > π 2 4 we have studied, there is also a jump on the imaginary axis in (4. 16), such that the local parametrix is constructed in terms of the Gamma functions in (4.67). However, we have not found a suitable parametrix near 0 when c < π 2 4 .…”

“…In the present paper, we will focus on the "saturated region-band-void" case when c > c cr and study asymptotics of the corresponding orthogonal polynomials. In the literature, such kind of problems have been studied by many researchers; see, for example, [1,3,4,16]. In our case, the origin needs some careful treatments because it is the hard edge of the equilibrium measure µ 0 .…”

“…Since the work of Baik et al [1], there has been a lot of work in the study of asymptotics of discrete orthogonal polynomials. For example, Wong and his co-workers derived global asymptotics of various polynomials, with finite lattice [4,15] and infinite lattice [16,20]. In the treatment of the so-called "band-saturated region" endpoints, Bleher and Liechty [2,3] made a major modification to the method in [1].…”