A Uniform Asymptotic Expansion for Krawtchouk Polynomials

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.…”

Asymptotic analysis of the Krawtchouk polynomials by the WKB method

“…He derived an approximation in terms of the Hermite polynomials (see also [36] for a similar formula). The general case with x, n = O(N) was investigated by Ismail and Simeonov [12] and a uniform asymptotic expansion was derived by Li and Wong [22], both using the saddle point method.…”

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].…”