Asymptotic expansion of the Tricomi-Carlitz polynomials and their zeros

The Tricomi-Carlitz polynomials f_n(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 f_n(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).© 2013 Elsevier B.V. All rights reserved.