Uniform Asymptotic Expansions for Charlier Polynomials

AbstractThe asymptotic behaviour of the Charlier polynomials C(a)n(x) as n→∞ is examined. These polynomials satisfy a discrete orthogonality relation and, unlike classical orthogonal polynomials, do not satisfy a second-order linear differential equation with respect to the independent variable x. As such, previous results on their asymptotic behaviour have been restricted to integral methods and consequently have been quite limited in their scope. In this paper a new approach is used, where the polynomials C(a)n(x) are not regarded as a functions of x with a as a parameter, but rather with the roles reversed via a second-order linear differential equation in which a is the (real or complex-valued) independent variable and x is a parameter. This equation has two turning points in the a plane which depend on x, and are either positive or complex conjugates, according to the values of x. Moreover, the turning points can coalesce with one another, or one with a singularity of the equation, for certain critical values of x. By using two general asymptotic theories of differential equations, one for intervals free of turning points and the other for intervals containing a double pole and a coalescing turning point, expansions are derived for C(a)n(x) involving either elementary functions or Bessel functions. Taken together, the results are uniformly valid for −∞<x<∞. In addition, in each case the expansions are uniformly valid for a lying in certain unbounded intervals, each of which contain [−{1−δ}n, {1−δ}n], where δ∈(0, 1) is an arbitrary constant