Strong asymptotics for Krawtchouk polynomials

Non-intersecting paths, random tilings and random matrices

“…The proof is based on sufficiently good asymptotic control of the Krawtchouk kernel, (2.9). For results on asymptotics of Krawtchouk polynomials see [35]. We state the needed result as a lemma, which we will prove later.…”

Discrete Analogues of the Laguerre Inequality

“…We outline the method leaving aside the details of calculations which are similar to these in [5], [6], and require a symbolic package, we used Mathematica. Asymptotic results obtained via a more classical approach can be found in [8], [14]. Let x 1 < ... < x k , be the zeros of P k (x).…”

“…The classical approach (see e.g. [8], [14]), heavily depends on the fact that the corresponding generating functions are of a rather simple form allowing asymptotic investigation. As far as we know the method of Laguerre type inequalities is the only one producing sharp explicit bounds in the discrete case.…”

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

“…Also, Lloyd's theorem [13,16] states that the existence of a perfect code in the Hamming metric corresponds to the Krawtchouk polynomials having integer zeros. Recently, there has been a considerable amount of interest in the asymptotics of the Krawtchouk polynomials, when the degree n grows to infinity; see [11,15,19].…”

“…Since the Krawtchouk polynomials do not satisfy a differential equation, most of the results in the literature are obtained by using the steepest descent method or the saddle point method for integrals, which come from the generating function in (1.1); see [11,15,19]. For more information about these classical integral methods, we refer to Wong [25].…”