Uniform Asymptotic Expansions for Meixner Polynomials

Meixner polynomials m n (x;β,c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are $j(x;\beta,c) = \frac{c^x(\beta)_x}{x!} \qquad \mbox{at}\quad x=0,1,2\ldots.$ Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. In this paper, we derive two infinite asymptotic expansions for m n (nα;β,c) as $n\to\infty$ . One holds uniformly for $0 < \epsilon\le \alpha\le 1+a$ , and the other holds uniformly for $1-b\le \alpha\le M < \infty$ , where a and b are two small positive quantities. Both expansions involve the parabolic cylinder function and its derivative. Our results include all five asymptotic formulas recently given by W. M. Y. Goh as special cases.