Asymptotic expansion of the Tricomi–Carlitz polynomials and their zeros

Abstract

The Tricomi–Carlitz polynomials fn(α)(x) are non-classical discrete orthogonal polynomials on the real line with respect to the step function whose jumps are dψ(α)(x)=(k+α)k−1e−kk!atx=xk=±(k+α)−1/2,k=0,1,2,…. In this paper, we derive an asymptotic expansion for fn(α)(t/ν) as n→∞, valid uniformly for bounded real t, where ν=n+2α−1/2 and α is a positive parameter. The validity for bounded t can be extended to unbounded t by using a sequence of rational functions introduced by Olde Daalhuis and Temme. The expansion involves the Airy functions and their derivatives. Error bounds are given for one-term and two-term approximations. Asymptotic formulas are also presented for the zeros of fn(α)(t/ν).