Asymptotic expansion of the Krawtchouk polynomials and their zeros (Q1880512)

The generating function \( (1-pw)^{ N-x}(1+qw)^x=\sum_{n=0}^\infty K_n^N(x;p,q) w^n \) and Cauchy's formula are used for obtaining an asymptotic expansion for large \(n\) for the Krawtchouk polynomials \(K_n^N(x;p,q)\). The expansion holds for fixed or bounded \(x\) and is uniformly for \(\mu=N/n \in [1,\infty)\). The main approximants are confluent hypergeometric functions. Asymptotic approximations are also derived for the zeros of \(K_n^N(x;p,q)\) for various cases depending on the values of \(p\), \(q\), and \(\mu\).